2025​​​‌Activity reportProject-TeamCARDAMOM‌

2.1 Scientific context and​​ challenges

The objective of​​​‌ CARDAMOM is to provide‌ improved analysis and design‌​‌ tools for engineering applications​​ involving fluid flows with​​​‌ moving fronts. In our‌ applications a front is‌​‌ either an actual material​​ interface, a boundary of​​​‌ the domain, or a‌ well identified transition region‌​‌ in which the flow​​ undergoes a change in​​​‌ its dominant macroscopic character‌. One example is‌​‌ the certification of wing​​ de-anti icing systems, involving​​​‌ the predictions of ice‌ formation and detachment, and‌​‌ of ice debris trajectories​​ to evaluate the risk​​​‌ of downstream impact on‌ aircraft components 85,‌​‌ 39. Another application,​​ relevant for space reentry,​​​‌ is the study of‌ transitional regimes in high‌​‌ altitude gas dynamics in​​ which extremely thin layers​​​‌ appear in the flow‌ which cannot be analysed‌​‌ with classical continuous models​​ (Navier-Stokes equations) used by​​​‌ engineers 46, 64‌. A classical example‌​‌ relevant in coastal engineering​​ is free surface flows.​​​‌ The free surface itself‌ is a material interface,‌​‌ but we can identify​​ also other fronts as​​​‌ e.g. the flooding line‌ (wet/dry transition) or the‌​‌ transition between propagating and​​ breaking waves, across which​​​‌ relevance of dissipation and‌ vorticity changes dramatically 47‌​‌. For wave energies,​​​‌ as well as for​ aquifers, the transition between​‌ free surface and congested​​ flows (below a solid​​​‌ surface) is another example​ 56. Other similar​‌ examples exist in geophysics,​​ astrophysics, aeronautic and aerospace​​​‌ engineering, civil engineering, energy​ engineering, material engineering, etc.​‌

In all cases, computationally​​ affordable, fast, and accurate​​​‌ numerical modelling is essential​ to allow reliable predictions​‌ in early stages of​​ the design/analysis 88.​​​‌ Such computational models are​ also needed for simulations​‌ over very long times,​​ especially if changes in​​​‌ many variable input parameters​ need to be investigated.​‌

To achieve this goal​​ one needs to have​​​‌ a physically relevant Partial​ Differential Equation (PDE) model,​‌ which can be treated​​ numerically efficiently and accurately,​​​‌ which means possibly with​ some adaptive numerical technique​‌ allowing to minimize the​​ computational effort. To this​​​‌ end, the dynamics of​ some of the fronts​‌ can be modelled by​​ appropriate asymptotic/homogenised PDEs, while​​​‌ other interfaces are explicitly​ described. Even in the​‌ best of circumstances in​​ all practical applications the​​​‌ reliability of the numerical​ predictions is limited by​‌ the intrinsic uncertainty on​​ the operational conditions (e.g.​​​‌ boundary/initial conditions, geometry, etc.).​ To this aleatory uncertainty​‌ we must add the​​ structural epistemic uncertainty related​​​‌ possibly to the use​ of approximate PDE models.​‌ Besides the limited validity​​ of the derivation assumptions,​​​‌ these models are often​ calibrated/validated with experimental data​‌ which is itself subject​​ to errors and post-processing​​​‌ procedures (filtering, averaging, etc​ ..) 51, 77​‌. This is even​​ worse in complex flows​​​‌ for which measurements are​ difficult or impossible to​‌ plan or perform due​​ to the inherent exceptional​​​‌ character of the phenomenon​ (e.g. tsunami events), or​‌ technical issues and danger​​ (e.g. high temperature reentry​​​‌ flows, or combustion), or​ impracticality due to the​‌ time scales involved (e.g.​​ study of some new​​​‌ materials' micro-/meso- structure 52​). So the challenge​‌ is to construct computationally​​ affordable models robust under​​​‌ variability of input parameters​ due to uncertainties, certification/optimization,​‌ as well as coming​​ from modelling choices.

To​​​‌ face this challenge and​ provide new tools to​‌ accurately and robustly modelize​​ and certify engineering devices​​​‌ based on fluid flows​ with moving fronts, we​‌ propose a program mixing​​ scientific research in asymptotic​​​‌ PDE analysis, high order​ adaptive PDE discretizations and​‌ uncertainty quantification.

2.2 Our​​ approach and objectives

We​​​‌ propose a research program​ mixing asymptotic PDE modelling,​‌ high order adaptive discretizations,​​ and uncertainty quantification. In​​​‌ a standard approach a​ certification study can be​‌ described as a modelling​​ exercise involving two black​​​‌ boxes. The first box​ is the computational model​‌ itself, composed of: PDE​​ system, mesh generation/adaptation, and​​​‌ discretization of the PDE​ (numerical scheme). The second​‌ box is the main​​ robust certification loop which​​​‌ contains separate boxes involving​ the evaluation of the​‌ physical model, the post-processing​​ of the output, and​​​‌ the exploration of the​ spaces of physical and​‌ stochastic parameters (uncertainties). Many​​ interactions exist in this​​​‌ process. Exploiting these interactions​ could allow to tap​‌ as much as possible​​ into the potential of​​ high order methods 67​​​‌ such as e.g. h-,‌ p-, r- adaptation in‌​‌ the physical model w.r.t.​​ some parametric quantity/sensitivity non​​​‌ necessarily associated to the‌ solution's smoothness.

Our objective‌​‌ is to provide some​​ fundamental advances allowing to​​​‌ bring closer to the‌ operational level modern high‌​‌ order numerical techniques and​​ multi-fidelity certification and optimization​​​‌ algorithms, possibly using some‌ clever paradigm different from‌​‌ the 2-black box approaches​​ above, and involving tight​​​‌ interactions between all the‌ parts of the play:‌​‌ PDE modelling, numerical discretization​​ techniques, uncertainty quantification methods,​​​‌ mesh generation/adaptation methods, physical‌ model validation/calibration, etc. The‌​‌ initial composition of the​​ team provided a unique​​​‌ combination of skills covering‌ all the necessary topics‌​‌ allowing to explore such​​ an avenue. The questions​​​‌ that need to be‌ tackled can be organized‌​‌ in the following main​​ axes/scientific questions:

1. Continuous modelling:​​​‌ how to obtain the‌ PDE description most suited‌​‌ for a given application,​​ and make sure that​​​‌ on one hand its‌ structure embeds sufficiently the‌​‌ physics sudied, and on​​ the other the system​​​‌ is in a form‌ suitable for efficient numerical‌​‌ discretization ?
2. Higher order​​ adaptive discretization: what are​​​‌ the relations between PDE‌ model accuracy (e.g. asymptotic‌​‌ error), PDE constraints (e.g.​​ entropy inequalities, particular steady​​​‌ states, etc) and the‌ scheme consistency ? how‌​‌ to account for additional​​ constraints in the scheme​​​‌ ?
3. Parameter uncertainty and‌ robust modelling: how to‌​‌ properly account when build​​ models on one hand​​​‌ for the variability of‌ physical states defining a‌​‌ process in realistic environments,​​ and on the other​​​‌ of data possibly available‌ for the process in‌​‌ consideration ? is it​​ possible to couple the​​​‌ sampling in the space‌ of parameters with the‌​‌ approximation in physical space​​ ?

These themes are​​​‌ discussed in the following‌ sections together with some‌​‌ challenges specific to the​​ engineering applications considered:

• Aeronautics​​​‌ and aerospace engineering (de-anti‌ icing systems, space re-entry,‌​‌ complex materials);
• Coastal engineering​​ (coastal protection, hazard assessment​​​‌ etc.);
• Energy engineering with‌ a focus on wave‌​‌ energy conversion
• Large scale​​ models on manifolds with​​​‌ a focus on geophysics‌ and some applications in‌​‌ astrophysics and relativity.

3.1 Continuous​​​‌ and discrete asymptotic modelling‌

In many of the‌​‌ applications we consider intermediate​​ fidelity models can be​​​‌ derived using an asymptotic‌ expansion for the relevant‌​‌ scale resolving PDEs, possibly​​ combined with some form​​​‌ of homogenization or averaging.‌ The resulting systems of‌​‌ PDEs are often very​​ complex. One of the​​​‌ main challenges is to‌ characterize the underlying structure‌​‌ of such systems: possible​​ conservation laws embedded; additional​​​‌ constraints related to consistency‌ with particular physical states‌​‌ (exact solutions), or to​​ stability (entropy/energy dissipation); etc.​​​‌ A question of paramount‌ importance in practical applications‌​‌ is also the formulation​​ of the boundary conditions.​​​‌ The understanding of these‌ properties is necessary for‌​‌ any new model. Moreover,​​ different forms of the​​​‌ PDE may be better‌ suited to enforce some‌​‌ of these properties at​​ the numerical level.

Another​​​‌ issue when working with‌ asymptotic approximations is that‌​‌ of closure. Indeed, important​​​‌ physical phenomena may be​ unaccounted for either due​‌ to some initial modelling​​ assumptions, or because they​​​‌ involve scales much smaller​ than those modelled. A​‌ typical example is wave​​ breaking in some depth​​​‌ averaged models. Another, relevant​ for our work, is​‌ the appropriate prediction of​​ heat fluxes in turbulent​​​‌ flows.

So our main​ activities on this axis​‌ can be classified according​​ to three main questions:​​​‌

• what is the structure​ of the PDE model​‌ (exact solutions, stability and​​ algebraic or differential constraints​​​‌ embedded, boundary conditions) ?​
• what is the form​‌ of the model better​​ suited to reproduce numerically​​​‌ certain constraints ?
• how​ to embed and design​‌ closure laws for relevant​​ phenomena not modelled by​​​‌ the main PDE ?​

3.2 High order discretizations​‌ on moving adaptive meshes​​

The efficient and robust​​​‌ discretization of complex PDEs​ is a classical and​‌ widespread research subject. The​​ notion of efficiency is​​​‌ in general related to​ the combination of high​‌ order of accuracy and​​ of some adaptation strategy​​​‌ based on an appropriate​ model of the error​‌ 79, 87.​​

This strategy is of​​​‌ course also part of​ our work. However, we​‌ are convinced that a​​ more effective path to​​​‌ obtain effective discretizations consists​ in exploiting the knowledge​‌ of the PDE structure,​​ embedding as much as​​​‌ possible the PDE structure​ in the discrete equations.​‌ This is related to​​ the notion of enhanced​​​‌ consistency that goes in​ the direction of what​‌ is today often referred​​ to as constraint or​​​‌ property preserving discretizations. For​ the type of PDE​‌ systems of our interest,​​ the properties which are​​​‌ of paramount importance to​ be controlled are for​‌ example: the balance between​​ flux divergence and forcing​​​‌ terms (so called well​ balanced of C-property 42​‌, 76) and​​ the preservation of some​​​‌ specific steady states; the​ correct reproduction of the​‌ dispersion relation of the​​ system, especially but not​​​‌ only for dispersive wave​ propagation; the preservation of​‌ some algebraic constraints, typically​​ the non-negativity of some​​​‌ thermodynamic quantities; the respect​ of a discrete entropy/energy​‌ equality or inequality (for​​ stability); the strong consistency​​​‌ with some asymptotic limit​ of the PDE (AP​‌ property); etc.

A fundamental​​ issue is the efficient​​​‌ and accurate treatment of​ boundary and interface conditions.​‌ The idea is to​​ have some approach which​​​‌ tolerates the use of​ non-conformal meshes, which is​‌ genuinely high order, and​​ compatible with adaptation, and​​​‌ of course conformal meshing​ of the boundary/discontinuity. Techniques​‌ allowing the control of​​ the geometrical error due​​​‌ to non-conformity is required.​ For discontinuities, this also​‌ requires an ad-hoc treatment​​ of the jump condition.​​​‌ For wall boundaries, initial​ work using penalization has​‌ been done in CARDAMOM​​ in the past 35​​​‌, 72. On​ Cartesian meshes several techniques​‌ exist to control the​​ consistency order based on​​​‌ extrapolation/interpolation, or adaptive methods​ (cf e.g.80,​‌ 71, 37,​​ 50, 58,​​​‌ 86 and references therein).​ For discontinuities, we can​‌ learn from fitting techniques​​ 43, and from​​ some past work by​​​‌ Prof. Glimm and co-workers‌ 49.

For efficiency,‌​‌ mesh adaptation plays a​​ major role. Mesh size​​​‌ adaptation based on both‌ deformation, r-adaptation, or remeshing‌​‌ h-adaptation, can be designed​​ based on some error​​​‌ model representative. For unsteady‌ flows, the capability to‌​‌ use moving meshes becomes​​ necessary. A major question​​​‌ is what conservation means‌ when geometry is modified.‌​‌ The geometrical conservation (GCL)​​ needs of course to​​​‌ be added to the‌ list of constraints to‌​‌ be accounted for 81​​, 68.

3.3​​​‌ Applications in physics and‌ engineering

As already mentioned,‌​‌ our focus is on​​ four main classes of​​​‌ problems:

• Aeronautics and aerospace‌ engineering (de-anti icing systems,‌​‌ space re-entry, complex materials)​​
• Coastal engineering (coastal protection,​​​‌ hazard assessment etc.)
• Energy‌ engineering with a focus‌​‌ on wave energy conversion​​
• Large scale models on​​​‌ manifolds with a focus‌ on geophysics (and possibly‌​‌ astrophysics).

There are several​​ common aspects. One is​​​‌ the use of asymptotic‌ vertically averaged approximations to‌​‌ produce efficient application-Taylored PDE​​ models. Another common point​​​‌ is the construction of‌ possibly high order constraint/property‌​‌ preserving numerical approximations. This​​ entails the characterization of​​​‌ the underlying PDE models‌ with a set of‌​‌ embedded properties, which go​​ from classical conservation, to​​​‌ exact solutions (steady or‌ moving), to the preservation‌​‌ of differential operators, to​​ the thermodynamic admissibility (non-negativity,​​​‌ preservation of physical bounds).‌ For all applications, the‌​‌ investigation of the parameter​​ dependence of the results​​​‌ will take several forms‌ from sensitivity analyses, to‌​‌ classical parametric studies to​​ understand physical processes, to​​​‌ approximation in parameter space‌ in the framework of‌​‌ hybrid PDE-meta-/reduced-order models.

4.1 De-anti​​​‌ icing systems

Impact of‌ large ice debris on‌​‌ downstream aerodynamic surfaces and​​ ingestion by aft mounted​​​‌ engines must be considered‌ during the aircraft certification‌​‌ process. It is typically​​ the result of ice​​​‌ accumulation on unprotected surfaces,‌ ice accretions downstream of‌​‌ ice protected areas, or​​ ice growth on surfaces​​​‌ due to delayed activation‌ of ice protection systems‌​‌ (IPS) or IPS failure.​​ This raises the need​​​‌ for accurate ice trajectory‌ simulation tools to support‌​‌ pre-design, design and certification​​ phases while improving cost​​​‌ efficiency. Present ice trajectory‌ simulation tools have limited‌​‌ capabilities due to the​​ lack of appropriate experimental​​​‌ aerodynamic force and moment‌ data for ice fragments‌​‌ and the large number​​ of variables that can​​​‌ affect the trajectories of‌ ice particles in the‌​‌ aircraft flow field like​​ the shape, size, mass,​​​‌ initial velocity, shedding location,‌ etc... There are generally‌​‌ two types of model​​ used to track shed​​​‌ ice pieces. The first‌ type of model makes‌​‌ the assumption that ice​​ pieces do not significantly​​​‌ affect the flow. The‌ second type of model‌​‌ intends to take into​​ account ice pieces interacting​​​‌ with the flow. We‌ are concerned with the‌​‌ second type of models,​​ involving fully coupled time-accurate​​​‌ aerodynamic and flight mechanics‌ simulations, and thus requiring‌​‌ the use of high​​ efficiency adaptive tools, and​​​‌ possibly tools allowing to‌ easily track moving objects‌​‌ in the flow. We​​​‌ will in particular pursue​ and enhance our initial​‌ work based on adaptive​​ immerse boundary capturing of​​​‌ moving ice debris, whose​ movements are computed using​‌ basic mechanical laws.

In​​ 40 it has been​​​‌ proposed to model ice​ shedding trajectories by an​‌ innovative paradigm that is​​ based on CArtesian grids,​​​‌ PEnalization and LEvel Sets​ (LESCAPE code). Our objective​‌ is to use the​​ potential of high order​​​‌ unstructured mesh adaptation and​ immersed boundary techniques to​‌ provide a geometrically flexible​​ extension of this idea.​​​‌ These activities will be​ linked to the development​‌ of efficient mesh adaptation​​ and time stepping techniques​​​‌ for time dependent flows,​ and their coupling with​‌ the immersed boundary methods​​ we started developing in​​​‌ the FP7 EU project​ STORM 35, 72​‌. In these methods​​ we compensate for the​​​‌ error at solid walls​ introduced by the penalization​‌ by using anisotropic mesh​​ adaptation 55, 69​​​‌, 70. From​ the numerical point of​‌ view one of the​​ major challenges is to​​​‌ guarantee efficiency and accuracy​ of the time stepping​‌ in presence of highly​​ stretched adaptive and moving​​​‌ meshes. Semi-implicit, locally implicit,​ multi-level, and split discretizations​‌ will be explored to​​ this end.

Besides the​​​‌ numerical aspects, we will​ deal with modelling challenges.​‌ One source of complexity​​ is the initial conditions​​​‌ which are essential to​ compute ice shedding trajectories.​‌ It is thus extremely​​ important to understand the​​​‌ mechanisms of ice release.​ With the development of​‌ next generations of engines​​ and aircraft, there is​​​‌ a crucial need to​ better assess and predict​‌ icing aspects early in​​ design phases and identify​​​‌ breakthrough technologies for ice​ protection systems compatible with​‌ future architectures. When a​​ thermal ice protection system​​​‌ is activated, it melts​ a part of the​‌ ice in contact with​​ the surface, creating a​​​‌ liquid water film and​ therefore lowering ability of​‌ the ice block to​​ adhere to the surface.​​​‌ The aerodynamic forces are​ then able to detach​‌ the ice block from​​ the surface 41.​​​‌ In order to assess​ the performance of such​‌ a system, it is​​ essential to understand the​​​‌ mechanisms by which the​ aerodynamic forces manage to​‌ detach the ice. The​​ current state of the​​​‌ art in icing codes​ is an empirical criterion.​‌ However such an empirical​​ criterion is unsatisfactory. Following​​​‌ the early work of​ 45, 39 we​‌ will develop appropriate asymptotic​​ PDE approximations to describe​​​‌ the water runoff on​ the wing surface, also​‌ accounting for phase change,​​ thus allowing to describe​​​‌ the ice formation and​ possibly rupture and detachment.​‌ These models will constitute​​ closures for aerodynamics/RANS and​​​‌ URANS simulations in the​ form of PDE wall​‌ models, or modified boundary​​ conditions.

In addition to​​​‌ this, several sources of​ uncertainties are associated to​‌ the ice geometry, size,​​ orientation and the shedding​​​‌ location. In very few​ papers 74, some​‌ sensitivity analysis based on​​ Monte Carlo method have​​​‌ been conducted to take​ into account the uncertainties​‌ of the initial conditions​​ and the chaotic nature​​ of the ice particle​​​‌ motion. We aim to‌ propose some systematic approach‌​‌ to handle every source​​ of uncertainty in an​​​‌ efficient way relying on‌ some state-of-art techniques developed‌​‌ in the Team. In​​ particular, we will perform​​​‌ an uncertainty propagation of‌ some uncertainties on the‌​‌ initial conditions (position, orientation,​​ velocity,...) through a low-fidelity​​​‌ model in order to‌ get statistics of a‌​‌ multitude of particle tracks.​​ This study will be​​​‌ done in collaboration with‌ ETS (Ecole de Technologies‌​‌ Supérieure, Canada). The long-term​​ objective is to produce​​​‌ footprint maps and to‌ analyse the sensitivity of‌​‌ the models developed.

4.2​​ Modelling of wave energy​​​‌ converters

Wave energy conversion‌ is an emerging sector‌​‌ in energy engineering. The​​ design of new and​​​‌ efficient Wave Energy Converters‌ (WECs) is thus a‌​‌ crucial activity. As pointed​​ out by Weber 88​​​‌, it is more‌ economical to raise the‌​‌ technology performance level (TPL)​​ of a wave energy​​​‌ converter concept at low‌ technology readiness level (TRL).‌​‌ Such a development path​​ puts a greater demand​​​‌ on the numerical methods‌ used.

Our previous work‌​‌ 5644 has shown​​ the potential of depth-averaged​​​‌ models for simulating wave‌ energy devices. The approach‌​‌ followed so far relies​​ on an explicit coupling​​​‌ of the different domains‌ involving the flow under‌​‌ the structure and the​​ free surface region. This​​​‌ approach has the advantage‌ to need efficient solvers‌​‌ of well-known system of​​ equations (compressible and incompressible​​​‌ flow). However, the transmission‌ condition between this two‌​‌ regimes is now always​​ well understood, depending on​​​‌ the underlying PDE models.‌ Moreover, several sources of‌​‌ numerical instabilities exist because​​ of the different nature​​​‌ of the regions involved‌ (compressible/incompressible). A different approach‌​‌ is proposed in 62​​, 61, and​​​‌ will be pursued in‌ the coming years. The‌​‌ idea is to solve​​ a unique model in​​​‌ the whole computational domain,‌ with the effect of‌​‌ the structure being accounted​​ for by means of​​​‌ an appropriate pressure variable‌ playing the role of‌​‌ a Lagrange multiplier. Our​​ numerical developments will be​​​‌ performed within the parallel‌ platform GeoFun, based‌​‌ on the Aerosol library.​​ In order to simulate​​​‌ the dynamic of the‌ floating structures, we will‌​‌ consider the coupling with​​ the open source code​​​‌ Chrono, an external‌ code specialized in the‌​‌ resolution of the rigid​​ body dynamics. The coupling​​​‌ is still under development.‌ In parallel, we will‌​‌ add closure for other​​ complex physical effects as​​​‌ e.g. the modelling of‌ air pocket trapped under‌​‌ the structures. Several industrial​​ processes (SeaTurns,​​​‌ Hace...) are based‌ on chamber compressing air‌​‌ inside by the movement​​ of the water surface.​​​‌ This strategy has the‌ advantage of taking the‌​‌ turbines for energy production​​ out of the water.​​​‌ The strategy is based‌ on a polytropic modelling‌​‌ of the gas dynamics​​ taking into account merging​​​‌ and splitting of the‌ pockets, without a major‌​‌ impact on the efficiency​​ of the simulation (robustness​​​‌ and numerical cost). This‌ works benefits of the‌​‌ associated team LARME with​​​‌ RISE (C. Eskilson).

4.3​ Coastal and civil engineering​‌

Our objective is to​​ bridge the gap between​​​‌ the development of high​ order adaptive methods, which​‌ has mainly been performed​​ in the industrial context​​​‌ and environmental applications, with​ particular attention to coastal​‌ and hydraulic engineering. We​​ want to provide tools​​​‌ for adaptive non-linear modelling​ at large and intermediate​‌ scales (near shore, estuarine​​ and river hydrodynamics). We​​​‌ will develop multi-scale adaptive​ models for free surface​‌ hydrodynamics. Beside the models​​ and codes themselves, based​​​‌ on the most advanced​ numerics we will develop​‌ during this project, we​​ want to provide sufficient​​​‌ know how to control,​ adapt and optimize these​‌ tools.

We will focus​​ our effort in the​​​‌ understanding of the interactions​ between asymptotic approximation and​‌ numerical approximation. This is​​ extremely important in several​​​‌ ways. An example is​ the capability of a​‌ numerical model to handle​​ highly dispersive wave propagation.​​​‌ This is usually done​ by high accuracy asymptotic​‌ PDE expansions of by​​ means of multilayer models.​​​‌ In the first case,​ there is an issue​‌ with the constraints on​​ the numerical approximation. Investigations​​​‌ of appropriated error models​ for adaptivity in the​‌ horizontal may permit to​​ alleviate some of these​​​‌ constraints, allowing a reasonable​ use of lower order​‌ discretizations. Concerning multi-layer models,​​ we plan can use​​​‌ results concerning the relations​ between vertical asymptotic expansions​‌ and truncation/approximation error to​​ improve the models by​​​‌ some adaptive approach.

Another​ important aspect which is​‌ not understood well enough​​ at the moment is​​​‌ the role of dissipation​ in the evolution of​‌ the free surface dynamics,​​ and of course in​​​‌ wave breaking regions. There​ are several examples of​‌ breaking closure, going from​​ algebraic and PDE-based eddy​​​‌ viscosity methods 66,​ 78, 73,​‌ 54, to hybrid​​ methods coupling dispersive PDEs​​​‌ with hyperbolic ones, and​ trying to mimic wave​‌ breaking with travelling bores​​ 84, 83,​​​‌ 82, 65,​ 57. In both​‌ cases, numerical dissipation plays​​ an important role and​​​‌ the activation or not​ of the breaking closure,​‌ as well as on​​ the establishment of stationary​​​‌ travelling profiles, or on​ the appearance of solitary​‌ waves. These aspects are​​ related to the notion​​​‌ of numerical dissipation, and​ to its impact on​‌ the resulting numerical solutions.​​ These elements must be​​​‌ clarified to allow full​ control of adaptive techniques​‌ for the models used​​ in this type of​​​‌ applications.

A fundamental issue​ that needs to be​‌ addressed is the proper​​ discrete formulation of the​​​‌ boundary conditions for dispersive​ wave approximations. These conditions​‌ play of course a​​ critical role in applications​​​‌ and remain an open​ problem for most Boussinesq​‌ models.

4.4 Geophysics and​​ astrophysics

This is work​​​‌ is related to large​ scale simulations requiring the​‌ solution of PDEs on​​ manifolds. Examples are tsunami​​​‌ simulations, as those performed​ in the past in​‌ the TANDEM project led​​ by CEA 63,​​​‌ as well as some​ applications considered in the​‌ ANR LAGOON for climate​​ change. In the framework​​ of the MSCA project​​​‌ SuPerMan we started to‌ look into applications in‌​‌ astrophysics which also involve​​ similar issues. The idea​​​‌ is to consider both‌ coordinate changes related to‌​‌ mesh movement, and in​​ ALE formulations, as well​​​‌ as genuinely curved frames‌ of reference 59,‌​‌ and combinations of both​​ when for example considered​​​‌ mesh movement and adaptation‌ in curvilinear coordinates 38‌​‌. Challenges are related​​ to the appropriate PDE​​​‌ formulation, and the respect‌ of continuous constraints at‌​‌ the discrete level.

The​​ objective here is to​​​‌ devise the most appropriate‌ manifold representation, and formulate‌​‌ the PDE system in​​ the appropriate way allowing​​​‌ to embed as many‌ continuous constraints as possible‌​‌ (well balancing, energy conservation,​​ positivity preservation, etc). Embedding​​​‌ the ALE mapping will‌ be necessary to envisage‌​‌ adaptive strategies, improving on​​ 38 and 60.​​​‌

5.1 Awards

The​​ MSCA proposal COPERNICUS, submitted​​​‌ by Laura del Río‌ Martín, has been accepted‌​‌ and funded. The project​​ official start has been​​​‌ set to February 2026.‌

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6.1 Latest software​​​‌ developments

7.1 Structure preserving‌ numerical methods for evolutionary‌​‌ PDEs

Participants: Firas Dhaouadi​​, Laura Del Río​​​‌ Martín, Maria Kazolea‌, Martin Parisot,‌​‌ Mario Ricchiuto.

• Corresponding​​ member: Mario Ricchiuto

In​​​‌ 2025, we have further‌ pursued the research on‌​‌ novel numerical techniques for​​ hyperbolic PDEs, with a​​​‌ strong focus on the‌ design of stationarity preserving‌​‌ methods, and genuinely multi-dimensional​​ discrete frameworks. The first​​​‌ property is related to‌ the ability of a‌​‌ method to embed discrete​​ approximations of truly multidimensional​​​‌ arbitrary stationary states. In‌ one space dimension, this‌​‌ property is related to​​ the so-called well balanced​​​‌ property, of which it‌ is essentially a fully‌​‌ discrete version. In other​​ words the methods we​​​‌ design preserve much more‌ general stationary states than‌​‌ classical well balanced methods,​​ but only an approximate/discrete​​​‌ form. In multiple space‌ dimension this property cannot‌​‌ be achieved when using​​ classical numerical methods based​​​‌ on one-dimensional Riemann solvers.‌ The development of genuinely‌​‌ multidimensional methods sets our​​ research path in the​​​‌ wake of the is‌ a fundamental and rich‌​‌ development dating back to​​ the early work by​​​‌ P.L. Roe, B. van‌ Leer, H. Deconinck, and‌​‌ may others. We have​​ written a thorough review​​​‌ on this aspect in‌ 4. This work‌​‌ involves many collaborations, in​​ particular with U. Paris​​​‌ Cité/LJLL (M. Ciallella), CNRS‌ (W. Barsukow), U. Roma‌​‌ La Sapienza (D. Torlo),​​ U. Ziurich (R. Abgrall,​​​‌ Y. Liu), U. of‌ Verona (E. Gaburro) and‌​‌ U. of Trento (M.​​ Dumbser).

This year, the​​​‌ framework introduced in 6‌ to design stationarity preserving‌​‌ and involution preserving stabilized​​ finite elements for homogenous​​​‌ linear hyperbolic PDEs has‌ been extended to linear‌​‌ hyperbolic systems balance laws​​​‌ with very general source​ terms in 27.​‌ The equations considered possess​​ many multi-dimensional non-trivial steady​​​‌ states, which result from​ the equilibrium between derivatives​‌ of the unknowns in​​ different directions, and the​​​‌ sources. Standard numerical methods​ include stabilization terms which​‌ are incompatible with such​​ multi-dimensional solutions. This manifests​​​‌ itself in a diffusion​ of states that are​‌ supposed to remain stationary.​​ In this novel “global​​​‌ flux quadrature” formulation, we​ introduce a transverse integration​‌ of each component of​​ the conservative flux, and​​​‌ a cell integration of​ the source terms. This​‌ leads to the use​​ of a so-called global​​​‌ flux, which can also​ be interpreted as a​‌ pointwise integral of the​​ evolution operator over mesh​​​‌ sub-cells. All spatial derivatives​ and the sources are​‌ thus treated simultaneously. On​​ Cartesian meshes, these methods​​​‌ are stationarity preserving. Additionally,​ when this formulation is​‌ combined with interpolation on​​ Gauss-Lobatto nodes, consistency results​​​‌ can be obtained for​ the discrete equilibrium states,​‌ showing super-convergence properties. The​​ numerical results confirm the​​​‌ theoretical predictions, and show​ the tremendous benefits of​‌ the new formulation.

A​​ first extension to non-linear​​​‌ problems has been performed​ in 5 (accepted on​‌ J.Comput.Phys.). In this case​​ we use the same​​​‌ ideas in the context​ of Finite Volume methods.​‌ Classical Finite Volume schemes​​ for multi-dimensional problems include​​​‌ stabilization (e.g. via a​ Riemann solver), that is​‌ derived by considering several​​ one-dimensional problems in different​​​‌ directions. Such methods therefore​ ignore a possibly existing​‌ balance of contributions coming​​ from different directions, such​​​‌ as the one characterizing​ multi-dimensional stationary states. Instead​‌ of being preserved, they​​ are usually diffused away​​​‌ by such methods. Stationarity​ preserving methods use a​‌ better suited stabilization term​​ that vanishes at the​​​‌ stationary state, allowing the​ method to preserve it.​‌ This work presents a​​ general approach to stationarity​​​‌ preserving Finite Volume methods​ for nonlinear conservation/balance laws.​‌ As the work done​​ in the finite element​​​‌ setting, the new schemes​ use a multi-dimensional stationarity​‌ preserving quadrature strategy that​​ allows to naturally introduce​​​‌ genuinely multi-dimensional numerical fluxes.​ In particular, instead of​‌ using face fluxes with​​ two arguments (left and​​​‌ right state), we are​ led to introduce numerical​‌ fluxes at cell corners​​ which are a function​​​‌ of multiple states (the​ number of corner neighbours​‌ of a cell). This​​ is a novel setting​​​‌ which we plan to​ explore in depth in​‌ the future. The new​​ methods are shown to​​​‌ significantly outperform existing ones​ even if the latter​‌ are of higher order​​ of accuracy. This has​​​‌ been shown for linear​ acoustics, for the non-linear​‌ Euler equations and on​​ the nonlinear shallow water​​​‌ equations, with and without​ source terms, on both​‌ stationary and even on​​ non-stationary solutions.

The quadrature​​​‌ strategy used in 6​, 27, 5​‌ is currently limited to​​ Cartesian meshes. Based on​​​‌ a different strategy, multidimensional​ Riemann solvers on general​‌ unstructured polygonal Voronoi-like tessellations​​ have been introduced in​​​‌ 29. In the​ reference, a general finite​‌ volume formulation based on​​ corner multidimensional fluxes is​​ put forward. As in​​​‌ 5, numerical fluxes‌ depend on cell average‌​‌ states in all the​​ corner neighbours. Two complete​​​‌ Riemann solvers are proposed‌ in this setting. The‌​‌ first is a direct​​ extension of the Osher-Solomon​​​‌ Riemann solver to multiple‌ space dimensions. Here, the‌​‌ multidimensional numerical dissipation is​​ obtained by integrating the​​​‌ absolute value of the‌ flux Jacobians over a‌​‌ dual triangular mesh around​​ each node of the​​​‌ primal polygonal grid. The‌ required nodal gradient is‌​‌ then evaluated on a​​ local computational simplex involving​​​‌ the d+1 neighbours meeting‌ at each corner. The‌​‌ second method is a​​ genuinely multidimensional upwind flux.​​​‌ By introducing a fluctuation‌ form of finite volume‌​‌ methods with corner fluxes,​​ we show an equivalence​​​‌ with residual distribution schemes‌ (RD). This naturally allows‌​‌ us to construct genuinely​​ multidimensional upwind corner fluxes​​​‌ starting from existing and‌ well-known RD fluctuations. In‌​‌ particular, we explore the​​ use of corner fluxes​​​‌ obtained from the so-called‌ N scheme, i.e. the‌​‌ Roe's original optimal multidimensional​​ upwind advection scheme. Both​​​‌ methods use the full‌ eigenstructure of the underlying‌​‌ hyperbolic system and are​​ therefore complete by construction.​​​‌ A simple higher order‌ extension up to fourth‌​‌ order in space and​​ time is then introduced​​​‌ at the aid of‌ a CWENO reconstruction in‌​‌ space and an ADER​​ approach in time, leading​​​‌ to a family of‌ high order accurate fully-discrete‌​‌ one-step schemes based on​​ genuinely multidimensional Riemann solvers.​​​‌ We present applications of‌ our new numerical schemes‌​‌ to several test problems​​ for the compressible Euler​​​‌ equations of gas-dynamics. The‌ new methods were implemented‌​‌ in AleVoronoi, and the​​ numerical results show that​​​‌ the proposed schemes can‌ handle very strong shocks,‌​‌ they preserve certain stationary​​ shear waves exactly, and​​​‌ have much reduced numerical‌ dissipation compared to classical‌​‌ methods.

The schemes studied​​ in the last work​​​‌ do not have any‌ clear stationarity or involution‌​‌ preserving properties. However, besides​​ the shear preserving property​​​‌ and the reduced numerical‌ dissipation, they use CWENO‌​‌ polynomials to manage strong​​ discontinuities. On the contrary,​​​‌ the multidimensional discretizations in‌ 6, 27,‌​‌ 5 do not involve​​ any non-linear mechanism to​​​‌ control oscillations. A first‌ step to overcome this‌​‌ limitation has been done​​ in collaboration with Y.​​​‌ Liu and R. Abgrall‌ (U. of Zurich) in‌​‌ 25. In this​​ work, the stationarity preserving/global​​​‌ flux quadrature approach is‌ combined with the active‌​‌ flux method. To manage​​ discontinuities and preserve the​​​‌ non-negativity of densities (and‌ water depth for shallow‌​‌ water) we blend the​​ high-order stationarity preserving schemes​​​‌ with a first order‌ local Lax-Friedrichs one. The‌​‌ blending is done based​​ on previous work by​​​‌ 36. Further control‌ of oscillations is obtained‌​‌ using the oscillation eliminating​​ filter by 75.​​​‌ The compatibility with the‌ stationarity preserving property is‌​‌ accounted for in the​​ definition of the blending​​​‌ function. Numerical results confirm‌ the possibility of combining‌​‌ stationarity preservation with oscillation​​ free, solutions respecting the​​​‌ appropriate positivity constraints.

7.2‌ Efficient numerical modelling of‌​‌ free surface and non-linear​​​‌ waves

Participants: Firas Dhaouadi​, Maria Kazolea,​‌ Martin Parisot, Mario​​ Ricchiuto.

• Corresponding member:​​​‌ Maria Kazolea

We have​ continued our work on​‌ enhanced numerical modelling of​​ free surface and non-linear​​​‌ waves. This is a​ multi-faceted activity involving PDE​‌ modelling and analysis, numerical​​ analysis, validation and verification.​​​‌ The most challenging part​ of the work is​‌ the construction of efficient​​ macroscopic models which appropriately​​​‌ account for small scales,​ and in particular for​‌ both wave dispersion, and​​ dissipation effects, both of​​​‌ which may not be​ correctly modelled in the​‌ lowest order PDE approximations.​​

In this respect, the​​​‌ coupling of different models​ is often necessary. This​‌ requires the understanding of​​ the well posedness of​​​‌ the coupling both on​ the theoretical and numerical​‌ level. In 30 we​​ derive a new approach​​​‌ to analyze the coupling​ of linear Boussinesq and​‌ Saint-Venant shallow water wave​​ equations in the case​​​‌ where the interface remains​ at a constant position​‌ in space. We propose​​ a one-way coupling model​​​‌ as a reference, which​ allows us to obtain​‌ an analytical solution, prove​​ the well-posedness of the​​​‌ original coupled model and​ compute what we call​‌ the coupling error-a quantity​​ that depends solely on​​​‌ the choice of transmission​ conditions at the interface.​‌ We prove that this​​ coupling error is asymptotically​​​‌ small for a certain​ class of data and​‌ discuss its role as​​ a proxy for the​​​‌ full error with respect​ to the 3D water​‌ wave problem. Additionally, we​​ highlight that this error​​​‌ can be easily computed​ in other scenarios. We​‌ show that the coupling​​ error consists of reflected​​​‌ waves and argue that​ this explains some previously​‌ unexplained spurious oscillations reported​​ in the literature. Finally,​​​‌ we prove the well-posedness​ of the half-line linear​‌ Boussinesq problem.

A standard​​ technique to construct accurate​​​‌ and robust numerical approximations​ of multi-scale PDE models​‌ is to recast them​​ in hyperbolic form. The​​​‌ advantage of this approach​ is that standard high​‌ order methods for hyperbolic​​ PDEs can be used.​​​‌ In 28, we​ introduce novel approximate systems​‌ for dispersive and diffusive-dispersive​​ equations with nonlinear fluxes.​​​‌ For purely dispersive equations,​ we construct a first-order,​‌ strictly hyperbolic approximation. Local​​ well-posedness of smooth solutions​​​‌ is achieved by constructing​ a unique symmetrizer that​‌ applies to arbitrary smooth​​ fluxes. Under stronger conditions​​​‌ on the fluxes, we​ provide a strictly convex​‌ entropy for the hyperbolic​​ system that corresponds to​​​‌ the energy of the​ underlying dispersive equation. To​‌ approximate diffusive-dispersive equations, we​​ rely on a viscoelastic​​​‌ damped system that is​ compatible with the found​‌ entropy for the hyperbolic​​ approximation of the dispersive​​​‌ evolution. For the resulting​ hyperbolic-parabolic approximation, we provide​‌ a global well-posedness result.​​ Using the relative entropy​​​‌ framework 53, we​ prove that the solutions​‌ of the approximate systems​​ converge to solutions of​​​‌ the original equations. The​ structure of the new​‌ approximate systems allows to​​ apply standard numerical simulation​​​‌ methods from the field​ of hyperbolic balance laws.​‌ We confirm the convergence​​ of our approximations even​​ beyond the validity range​​​‌ of our theoretical findings‌ on set of test‌​‌ cases covering different target​​ equations. We show the​​​‌ applicability of the approach‌ for strong nonlinear effects‌​‌ leading to oscillating or​​ shock-layer-forming behaviour.

Hyperbolic relaxation​​​‌ techniques as the one‌ discussed in the previous‌​‌ contribution are widely used​​ to offset the cost​​​‌ of inverting time independent‌ operators. To give an‌​‌ example the Serre-Green-Naghdi (SGN)​​ equations provide a valuable​​​‌ framework for modelling fully‌ nonlinear and weakly dispersive‌​‌ shallow-water flows. However, their​​ standard formulation requires the​​​‌ inversion of an elliptic‌ PDE which can considerably‌​‌ increase the computational cost​​ compared to the Saint-Venant​​​‌ equations. To overcome this‌ difficulty, hyperbolic models (hSGN)‌​‌ have been proposed that​​ replace the elliptic operators​​​‌ with first-order hyperbolic formulations‌ augmented by relaxation terms,‌​‌ which recover the original​​ elliptic formulation in the​​​‌ stiff limit. Yet, as‌ the relaxation parameter λ‌​‌ increases, explicit schemes face​​ restrictive stability constraints that​​​‌ may offset these advantages.‌ To mitigate this limitation,‌​‌ in 32 we introduce​​ a semi-implicit (SI) integration​​​‌ strategy for the hSGN‌ system, where the stiff‌​‌ acoustic terms are treated​​ implicitly within an IMEX​​​‌ Runge-Kutta framework, while the‌ advective components remain explicit.‌​‌ The proposed approach mitigates​​ the CFL stability restriction​​​‌ and maintains dispersive accuracy‌ at a moderate computational‌​‌ cost. Numerical results confirm​​ that the combination of​​​‌ hyperbolization and semi-implicit time‌ integration may provide an‌​‌ efficient and accurate alternative​​ to both classical SGN​​​‌ and fully explicit hSGN‌ solvers.

Still on the‌​‌ numerical side, in 8​​ we have proposed enhance​​​‌ semi-implicit methods for wave‌ structure interaction problems with‌​‌ moving free surface, and​​ large relative motions between​​​‌ the fluid and moving‌ floating structures above water.‌​‌ As shown in our​​ previous works 44,​​​‌ 61, the governing‌ equations change type from‌​‌ hyperbolic below the free​​ surface to elliptic below​​​‌ the moving floating object‌ within the cells of‌​‌ the computational domain, the​​ horizontal motion may exhibit​​​‌ a numerical instability that‌ is not observed solely‌​‌ with the vertical heave​​ motion. When a ship​​​‌ enters a new cell,‌ the pressure at the‌​‌ bow increases and decreases​​ sharply, leading to oscillations​​​‌ that can create an‌ unphysical void below the‌​‌ vessel. After examining the​​ origin of the problem,​​​‌ several measures were taken‌ at the discrete level‌​‌ to reduce these spurious​​ oscillations. All of these​​​‌ modifications were effective in‌ controlling the oscillations, making‌​‌ numerical simulations with horizontal​​ motion of moving floating​​​‌ objects more robust, stable‌ and accurate. To enhance‌​‌ the range of application​​ of the model, we​​​‌ also include some additional‌ weakly dispersive terms. The‌​‌ same study is then​​ performed in presence of​​​‌ these non-hydrostatic effects, showing‌ a reliable and robust‌​‌ prediction of the interaction​​ of moving bodies interacting​​​‌ with hydrostatic and non-hydrostatic‌ (dispersive) waves.

In 33‌​‌, we presents a​​ novel approach for coupling​​​‌ a high-fidelity Navier-Stokes model‌ with an asymptotic Boussinesq-type‌​‌ model. The proposed methodology​​ aims to enhance the​​​‌ design and performance of‌ wave energy converters, thereby‌​‌ supporting the broader integration​​​‌ of wave energy in​ addressing climate change and​‌ energy resource challenges. The​​ numerical coupling strategy relies​​​‌ on overlapping subdomains with​ relaxation source terms transferred​‌ information from one model​​ to the other. This​​​‌ framework improves the accuracy​ of local wave-structure interaction​‌ studies while maintaining computational​​ efficiency over larger spatial​​​‌ domains. A series of​ case studies from the​‌ literature is carried out​​ to illustrate the relevance​​​‌ and effectiveness of the​ approach.

All developments of​‌ new numerical methods or​​ approximate models rely heavily​​​‌ on the availability of​ reliable, well documented, and​‌ relevant benchmarks. For this​​ reason, in 26 we​​​‌ have proposed a comprehensive​ list of stationary solutions​‌ of different shallow water​​ models. For each different​​​‌ case, we exhibit the​ form of the stationary​‌ solution and its conditions​​ of existence for any​​​‌ set of boundary conditions​ on monotonic, bumped shaped​‌ and hollow shaped topographies.​​ We find that in​​​‌ some cases and for​ supercritical flow at the​‌ inlet, two different stationary​​ solutions can fulfilled the​​​‌ same set of boundary​ conditions. The solutions with​‌ shocks on increasing topography​​ are sparsely documented and​​​‌ seem to be unstable.​ Eventually we consider a​‌ topography made of a​​ succession of N decreasing​​​‌ bumps and prove in​ the subcritical case that​‌ up to $2^{\mathrm{N}-1}$ solutions with​​​‌ discontinuities on decreasing topography​ only may coexist.

7.3​‌ Advanced models of heat​​ transfer and phase change​​​‌ phenomena

Participants: Héloïse Beaugendre​.

• Participants: Héloïse Beaugendre,​‌ Alexis Cas
• Corresponding member:​​ Héloïse Beaugendre

During atmospheric​​​‌ hypersonic re-entry, the heat​ distribution within the thermal​‌ protection system (TPS) is​​ dampened by the in-depth​​​‌ chemical degradation of materials​ - called pyrolysis -,​‌ and by a surface​​ physicochemical degradation - called​​​‌ ablation. The aim of​ Alexis Cas's PhD work​‌ work is to preserve​​ the mass and energy​​​‌ conservation and to investigate​ the numerical tools used​‌ during the pyrolysis-thermal coupling​​ of heat shield under​​​‌ deformations. First, an overview​ of macroscopic modeling of​‌ pyrolysis is done. Arrhenius​​ laws are employed for​​​‌ the modeling of density​ variation. Thermal expansion, swelling​‌ and shrinkage are taken​​ into account as a​​​‌ consequence of material degradation.​ This analysis explores a​‌ pyrolysis-thermal model that preserves​​ physical properties and a​​​‌ number of numerical methods,​ focusing on numerical conservation​‌ and method efficiency. Finally,​​ ablation and swelling test​​​‌ cases are studied, in​ order to validate and​‌ compare the numerical methods.​​ The simulation results are​​​‌ in reasonable agreement with​ reference data and experimental​‌ data. Some numerical methods​​ result in a trade-off​​​‌ between mass or energy​ conservation and a faster​‌ computational time.

7.4 High​​ order embedded and immersed​​​‌ boundary methods

Participants: Héloïse​ Beaugendre, Benjamin Constant​‌, Mario Ricchiuto.​​

• Corresponding member: Héloïse Beaugendre​​​‌

We studied several steady-state​ turbulent flow simulations of​‌ the NASA High Lift​​ Common Research Model on​​​‌ Cartesian grids at increasing​ angles of attack. The​‌ Reynolds Averaged Navier-Stokes equations​​ are solved with the​​​‌ one-equation Spalart-Allmaras model, while​ the presence of a​‌ geometry is accounted for​​ by an immersed boundary​​ method based on a​​​‌ ghost cell direct forcing‌ approach. A wall model‌​‌ is applied to deal​​ with the inherent impossibility​​​‌ of isotropic cells to‌ properly capture the near-wall‌​‌ flow physics at high​​ Reynolds number. This work​​​‌ validates the recent developments‌ brought to the Cartesian‌​‌ methodology based on a​​ geometric regularization of the​​​‌ forcing point location around‌ the immersed geometry. New‌​‌ developments are presented to​​ improve the pre- and​​​‌ post-processing for this type‌ of 3D applications. Comparisons‌​‌ are made with state-of-the-art​​ solutions from the reference​​​‌ literature on body-fitted meshes,‌ both structured and unstructured.‌​‌ A 2D multi-element airfoil​​ is also studied to​​​‌ facilitate the analysis of‌ the solutions obtained on‌​‌ the full aircraft in​​ high lift configuration. For​​​‌ higher angles of attack,‌ more detailed investigations are‌​‌ proposed to discuss the​​ physics closer to the​​​‌ stall phenomenon, especially near‌ the wing tip.

We‌​‌ also introduced a novel​​ data-driven wall modeling strategy​​​‌ for Reynolds-Averaged Navier-Stokes (RANS)‌ simulations. The method reformulates‌​‌ wall laws as a​​ Dirichlet-to-Neumann map applied at​​​‌ a specific height within‌ the boundary layer. This‌​‌ map is learned using​​ neural networks trained on​​​‌ data from wall-resolved RANS‌ simulations. While the approach‌​‌ shares similarities with the​​ work of Romanelli et​​​‌ al. (2023), it is‌ significantly faster, as it‌​‌ eliminates the need for​​ iterative resolution of the​​​‌ skin friction inherent in‌ the implicit relation ${\mathrm{u‌‌}}^{+}=f(y^{+},{\mathrm{p‌}}^{+})$. The‌ model's accuracy has also‌​‌ been improved through enhanced​​ treatment of the turbulent​​​‌ variables and by incorporating‌ additional input parameters that‌​‌ better describe the boundary​​ layer state. The method​​​‌ is demonstrated on attached‌ turbulent flows across various‌​‌ Reynolds numbers and wall​​ pressure gradients. After training​​​‌ the neural networks on‌ a subset of reference‌​‌ cases, their ability to​​ generalize to both familiar​​​‌ and unseen conditions is‌ evaluated. The model is‌​‌ further validated on a​​ completely different setup (an​​​‌ airfoil), where it is‌ compared to two existing‌​‌ analytical wall models, showing​​ better accuracy and robustness​​​‌ with respect to pressure‌ gradient conditions. This work‌​‌ also introduces an efficient​​ extrapolation-detection algorithm that could​​​‌ be used either to‌ ensure that the model‌​‌ is always applied within​​ its validity domain or​​​‌ to trigger an automated‌ adaptive learning strategy of‌​‌ the wall-law or, in​​ a zonal context, to​​​‌ select between wall-resolved/wall-modeled regions.‌ Overall, our method provides‌​‌ a new practical intermediatefidelity,​​ cost-effective framework that displays​​​‌ an attractive balance between‌ accuracy and computational efficiency,‌​‌ bridging the gap between​​ the more computationally intensive​​​‌ approach of Romanelli et‌ al. (2023) and the‌​‌ limited accuracy of conventional​​ analytical wall models.

7.5​​​‌ Adaptation techniques

Participants: Nicolas‌ Barral, Héloïse Beaugendre‌​‌, Luca Cirrottola,​​ Corentin Prigent, Mario​​​‌ Ricchiuto, Fabien Salmon‌, Ishak Tifouti.‌​‌

• Corresponding member: Nicolas Barral​​

MMG Consortium the anisotropic​​​‌ meshing library MMG is‌ supported by a consortium‌​‌ of industrial and academic​​ partners. After restarting in​​​‌ 2024, the consortium has‌ continued its activities of‌​‌ maintaining MMG, following the​​​‌ consortium roadmap. In September,​ a new version of​‌ the website was published,​​ with an extensive documentation​​​‌ of the code. In​ June 2025, the consortium​‌ met to update the​​ roadmap. Following this meeting,​​​‌ a new Python interface​ is being developped, while​‌ a continuous effort to​​ improuve robustness and reliability​​​‌ is ongoing.

Mesh adpatation​ for sea ice modelling.​‌ The collaboration with Institut​​ des Géosciences de l'Environnement,​​​‌ Grenoble, with Fabien Salmon​ 's. Three axes were​‌ followed. First, the performance​​ of parMMG2D was further​​​‌ assessed and improved. Second,​ a substantial effort to​‌ integrate MMG in the​​ state-of-the art sea-ice model​​​‌ NeXtSim, and merge the​ changes upstream in the​‌ main version of the​​ code. Finally, a theoretical​​​‌ study of error estimators​ for quasi isotropic and/or​‌ quasi uniform meshes was​​ carried out, resulting in​​​‌ new metrics to guide​ MMG in the context​‌ of sea-ice modelling.

Coupling​​ mesh adaptation with model​​​‌ reduction.

In 18,​ we propose a localized​‌ automated nonlinear model reduction​​ framework for rapid and​​​‌ reliable solution of parameterized​ shock-dominated flows. Our formulation​‌ exploits the adaptive procedure​​ of [Barral et al,​​​‌ JCP, 2024] and a​ clustering technique to devise​‌ a piecewise Lagrangian approximation​​ of the solution to​​​‌ the parametric problem: the​ application of clustering is​‌ designed to cope with​​ parameter-induced shock-topology changes that​​​‌ hinder the effectiveness of​ standard (monolithic) Lagrangian approximations.​‌ We rely on (i)​​ metric-based mesh adaptation to​​​‌ generate an accurate mesh​ for a range of​‌ parameters, (ii) parametric registration​​ for the computation of​​​‌ a bijection Φ that​ tracks moving features of​‌ the solution field, and​​ (iii) projection-based model reduction​​​‌ to rapidly and reliably​ estimate the mapped solution;​‌ finally, we develop (iv)​​ a clustering technique to​​​‌ partition the parameter domain​ in subregions where the​‌ shock topology does not​​ change. We present numerical​​​‌ investigations for a two-dimensional​ supersonic inviscid flow past​‌ a Gaussian bump to​​ illustrate the many features​​​‌ of the methodology: in​ more detail, we display​‌ the performance of three​​ clustering techniques, and we​​​‌ compare our approach with​ monolithic Lagrangian approximations.

Finally,​‌ a goal-oriented approach encompassing​​ MOR and mesh adaptation​​​‌ was designed. While goal-oriented​ methods are well established​‌ in mesh adaptation and​​ shape optimization, their use​​​‌ in model reduction is​ much less common. We​‌ developed an automated goal-oriented​​ adaptive multi-fidelity training algorithm​​​‌ that produces goal-oriented registration-based​ reduced models tailored to​‌ specific outputs of interest.​​ Two goal-oriented mesh adaptation​​​‌ strategies were considered: (i)​ an anisotropic approach based​‌ on an a priori​​ goal-oriented error estimator, and​​​‌ (ii) an isotropic approach​ derived from an a​‌ posteriori goal-oriented estimator. Both​​ methods were adapted to​​​‌ our parametric setting. We​ compared them in terms​‌ of mesh quality, solution​​ accuracy, and accuracy of​​​‌ the associated outputs, with​ a feature-based anisotropic mesh​‌ adaptation approach. As expected,​​ parametric solutions obtained with​​​‌ goal-oriented adapted meshes are​ less accurate than those​‌ built on feature-based meshes.​​ However, goal-oriented meshes are​​​‌ refined in significantly smaller​ regions of the computational​‌ domain, specifically those relevant​​ to the outputs of​​ interest, thereby reducing the​​​‌ relative output error compared‌ to the feature-based approach.‌​‌ For the supersonic flow​​ past a bump, we​​​‌ found that isotropic goal-oriented‌ meshes yield better accuracy‌​‌ in the output error,​​ whereas anisotropic goal-oriented meshes​​​‌ provide improved resolution of‌ the adjoint solution by‌​‌ following its spatial variations.​​

7.6 Modeling of flows​​​‌ in aquifers

Participants: Manon‌ Carreau, Martin Parisot‌​‌.

• Corresponding member: Martin​​ Parisot

In 9,​​​‌ we focus on the‌ formal derivation of a‌​‌ unified model for groundwater​​ and surface water flow,​​​‌ combining the shallow water‌ model and the Dupuit-Forchheimer‌​‌ model. The primary goal​​ of the model is​​​‌ to describe the dynamics‌ of water table, which‌​‌ represents the level below​​ which the medium is​​​‌ fully saturated, while disregarding‌ unsaturated regions. The unified‌​‌ shallow-water/Dupuit-Forchheimer model provides a​​ single set of equations​​​‌ across the domain. It‌ transitions to the shallow‌​‌ water model in areas​​ where the porous medium​​​‌ is absent and to‌ the Dupuit-Forchheimer model when‌​‌ the water table remains​​ below the surface. The​​​‌ derivation is carried out‌ in two steps: first,‌​‌ a vertical integration separates​​ the ground and surface​​​‌ media, followed by the‌ application of a low-permeability‌​‌ limit in the ground​​ layer. The model ensures​​​‌ conservation of both mass‌ and energy. A numerical‌​‌ scheme that preserves these​​ properties at the discrete​​​‌ level, along with several‌ numerical experiments, demonstrates the‌​‌ robustness and efficiency of​​ the approach.

8.1 Bilateral Contracts‌​‌

9.1 International research visitors​​​‌

9.2 European‌​‌ initiatives

9.3 National​​ initiatives

9.4 Regional​ initiatives

Participants: Maria Kazolea​‌, Martin Parisot,​​ Mario Ricchiuto.

10.1 Scientific events:​​​‌ organisation

• Maria Kazolea is‌ a member of the‌​‌ organizing Committee of the​​ international workshop Sunhype 2026​​​‌
• Mario Ricchiuto has co-organized‌ the workshop B'Waves25,‌​‌ focusing on various aspects​​ of wave breaking, with​​​‌ accent on free surface‌ waves. This is the‌​‌ sixth edition of the​​ B’Waves series started with​​​‌ a 2014 workshop aiming‌ at serving as a‌​‌ platform for interactive exchange​​ between scientists.

10.2 Journal​​​‌

10.3 Invited talks

• Martin‌​‌ Parisot has provided invited​​ talks at the following​​​‌ events
  • Climath, 25-27/11‌
  • Digest, 13/11
  • Hydraumath,‌​‌ 16-25/06
  • JEMP, 04-06/11​​
  • NumHyp, 09-13/06
  • Cimav​​​‌, 11-16/05
• Mario Ricchiuto‌ has provided invited talks‌​‌ or lectures at the​​ following events
  • Workshop on​​​‌ bedload and sedimentation processes:‌ experiments, modelling and numerical‌​‌ simulation, Sevilla (Spain),​​ January 2025
  • 1st PICASSO​​​‌ Workshop (France-Portugal-Spain), Malaga‌ (Spain), March 2025
  • SHARK-FV‌​‌ 2025 workshop, Minho (Portugal),​​ June 2025
  • Aria workshop​​​‌ 2025, Bidart (France),‌ September 2025
  • ETNA Workshop‌​‌ Efficient Time-Stepping Numerical Approaches​​ for PDEs, Catania​​​‌ (Italy), November 2025

10.4‌ Leadership within the scientific‌​‌ community

• H. Beaugendre is​​ member of the following​​​‌ committees
  • Scientific committee of‌ Forum des jeunes mathématiciennes‌​‌ et mathématiciens (JMM2025​​, november 2025, Bordeaux)​​​‌
  • Scientific Committee of CANUM‌ 2026
• Mario Ricchiuto is‌​‌ member of the following​​​‌ committees
  • Scientific Committee of​ the International Conference on​‌ Computational Fluid Dynamics –​​ ICCFD
  • Scientific Committee of​​​‌ the Terre et Énergies,​ Réseau Thématique 2166 du​‌ CNRS Mathématiques – RT​​ CNRS Terre et Énergies​​​‌

10.5 Scientific expertise

• H.​ Beaugendre has been an​‌ expert for the evaluation​​ of the research project​​​‌ PRF TRICEPS related to​ in-flight icing, in ONERA​‌ Toulouse, january 2025.
• Martin​​ Parisot served as an​​​‌ expert for the evaluation​ of the ECOS SUD​‌ CHILI research project, related​​ to collaboration between French​​​‌ and Chilean researchers.

10.6​ Research administration

• Since September​‌ 2024, H. Beaugendre has​​ been Director of Research,​​​‌ Innovation and Technology Transfer​ at ENSEIRB-MATMECA, Bordeaux INP.​‌
• Since January 2025, Maria​​ Kazolea has been serving​​​‌ as the Scientific Correspondent​ for the International Relations​‌ Department of the Inria​​ Center at the University​​​‌ of Bordeaux. As such​ she has contributed participated​‌ to the creation of​​ the software coPubli.​​​‌ CoPubli is a minimal​ software pipeline to extract,​‌ browse and geolocalise HAL​​ coauthors. This tool visualises​​​‌ Inria's international network and​ partnerships. CoPubli extract an​‌ Exel file containing for​​ each Inria Centre (respectively,​​​‌ Inria Team, or Inria​ Researcher) a list of​‌ foreign co-publishing researchers along​​ with their research institutions,​​​‌ city, and country, information​ that could not be​‌ easily extracted by HAL​​ without using data fusion​​​‌ and aggregation techniques 34​.
• Since January 2025,​‌ Martin Parisot has been​​ serving as referent of​​​‌ the Inria Center of​ Bordeaux for the Inria+Alumni​‌ community.

10.7 Teaching -​​ Supervision - Juries -​​​‌ Educational and pedagogical outreach​

10.8 Popularization

N.​​ Barral, M. Kazolea and​​​‌ M. Ricchiuto participated in‌ the article À Bordeaux,‌​‌ ces chercheurs qui simulent​​ numériquement des tsunamis of​​​‌ the regional newspaper SudOuest.‌

11.1‌​‌ Major publications

• 1 article​​R.Remi Abgrall,​​​‌ P.-H.Pierre-Henri Maire and‌ M.Mario Ricchiuto.‌​‌ Embedding General Conservation Constraints​​ in Discretizations of Hyperbolic​​​‌ Systems on Arbitrary Meshes:‌ A Multidimensional Framework.‌​‌Mathematical Models and Methods​​ in Applied SciencesDecember​​​‌ 2025HALDOI
• 2‌ articleW.Wasilij Barsukow‌​‌, M.Mario Ricchiuto​​ and D.Davide Torlo​​​‌. Structure preserving nodal‌ continuous Finite Elements via‌​‌ Global Flux quadrature.​​Numerical Methods for Partial​​​‌ Differential Equations411‌January 2025HAL
• 3‌​‌ articleM.Manon Carreau​​ and M.Martin Parisot​​​‌. A unified modeling‌ of underground-surface hydraulic processes‌​‌.Multiscale Modeling and​​ Simulation: A SIAM Interdisciplinary​​​‌ Journal241January‌ 2025, 68-98HAL‌​‌DOI

11.2 Publications of​​ the year

11.3 Cited publications

• 35​​​‌ articleR.Remi Abgrall​, H.Hélo\"ise Beaugendre​‌ and C.Cécile Dobrzynski​​. An immersed boundary​​​‌ method using unstructured anisotropic​ mesh adaptation combined with​‌ level-sets and penalization techniques​​.Journal of Computational​​​‌ Physics2572014,​ 83-101HALback to​‌ textback to text​​
• 36 articleR.R.​​​‌ Abgrall, M.M.​ Jiao, Y.Y.​‌ Liu and K.K.​​ Wu. Bound preserving​​​‌ Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme:​ one-dimensional case.Communications​‌ in Computational Physics39​​1Jan. 2026,​​​‌ 29-58URL: https://global-sci.com/index.php/cicp/article/view/23219DOI​back to text
• 37​‌ articleAn Immersed Boundary​​ Method with Formal Second-Order​​​‌ Accuracy and Reduced Numerical​ Viscosity.Journal of​‌ Computational Physics1602​​2000, 705-719URL:​​​‌ https://www.sciencedirect.com/science/article/pii/S0021999100964830DOIback to​ text
• 38 articleL.​‌Luca Arpaia and M.​​Mario Ricchiuto. Well​​​‌ balanced residual distribution for​ the ALE spherical shallow​‌ water equations on moving​​ adaptive meshes.Journal​​​‌ of Computational Physics405​2020, 109173HAL​‌DOIback to text​​back to text
• 39​​​‌ articleH.H. Beaugendre​ and F.F. Morency​‌. Development of a​​ Second Generation In-flight Icing​​ Simulation Code.Journal​​​‌ of Fluids Engineering128‌2006back to text‌​‌back to text
• 40​​ articleH.Hélo\"ise Beaugendre​​​‌, F.François Morency‌, F.Fédérico Gallizio‌​‌ and S.Sophie Laurens​​. Computation of Ice​​​‌ Shedding Trajectories Using Cartesian‌ Grids, Penalization, and Level‌​‌ Sets.Modelling and​​ Simulation in Engineering2011​​​‌2011, 1-15HAL‌back to text
• 41‌​‌ articleL.Lokman Bennani​​, P.Philippe Villedieu​​​‌, M.M Salaun‌ and P.Pierre Trontin‌​‌. Numerical simulation and​​ modeling of ice shedding:​​​‌ Process initiation.Computers‌ & Structures1422014‌​‌, 15--27back to​​ text
• 42 articleA.​​​‌A. Bermudez and M.‌ V.M.E. Vazquez Cendon‌​‌. Upwind Methods for​​ Hyperbolic Conservation Laws with​​​‌ Source Terms.Comput.‌ Fluids23581994‌​‌, 1049--1071back to​​ text
• 43 articleA.​​​‌A. Bonfiglioli, M.‌M. Grottadaurea, R.‌​‌R. Paciorri and F.​​F. Sabetta. An​​​‌ unstructured, three-dimensional, shock-fitting solver‌ for hypersonic flows.‌​‌Computers & Fluids73​​2013, 162 -​​​‌ 174back to text‌
• 44 articleU.Umberto‌​‌ Bosi, A. P.​​Allan P Engsig-Karup,​​​‌ C.Claes Eskilsson and‌ M.Mario Ricchiuto.‌​‌ A spectral/hp element depth-integrated​​ model for nonlinear wave-body​​​‌ interaction.Computer Methods‌ in Applied Mechanics and‌​‌ Engineering3482019,​​ 222-249HALDOIback​​​‌ to textback to‌ text
• 45 articleY.‌​‌Y. Bourgault, H.​​H. Beaugendre and W.​​​‌W.G. Habashi. Development‌ of a Shallow-Water icing‌​‌ model in FENSAP-ICE.​​Journal of Aircraft37​​​‌2000back to text‌
• 46 articleI. D.‌​‌I. D. Boyd,​​ G.G. Chen and​​​‌ G. V.G. V.‌ Candler. Predicting the‌​‌ Failure of the Continuum​​ Fluid Equations in Transitional​​​‌ Hypersonic Flow.Physics‌ of Fluids71995‌​‌, 210--219back to​​ text
• 47 articleM.​​​‌M. Brocchini. A‌ reasoned overview on Boussinesq-type‌​‌ models: the interplay between​​ physics, mathematics and numerics​​​‌.Proc. Royal Soc.‌ A4692014back‌​‌ to text
• 48 article​​R.Rémi Chassagne,​​​‌ A. G.Andrea Gilberto‌ Filippini, M.Mario‌​‌ Ricchiuto and P.Philippe​​ Bonneton. Dispersive and​​​‌ dispersive-like bores in channels‌ with sloping banks.‌​‌Journal of Fluid Mechanics​​8702019, 595--616​​​‌back to text
• 49‌ articleI.-L.I.-L Chern‌​‌, J.J Glimm​​, O.O Mcbryan​​​‌, B.B Plohr‌ and S.S Yaniv‌​‌. Front tracking for​​ gas dynamics.Journal​​​‌ of Computational Physics62‌11986, 83-110‌​‌URL: https://www.sciencedirect.com/science/article/pii/0021999186901014DOIback​​ to text
• 50 article​​​‌S.S. Clain,‌ D.D. Lopes and‌​‌ R.R.M.S. Pereira.​​ Very high-order Cartesian-grid finite​​​‌ difference method on arbitrary‌ geometries.Journal of‌​‌ Computational Physics4342021​​, 110217URL: https://www.sciencedirect.com/science/article/pii/S0021999121001121​​​‌DOIback to text‌
• 51 bookC. F.‌​‌Computational Fluid Dynamics Committee​​. Guide for the​​​‌ Verification and Validation of‌ Computational Fluid Dynamics Simulations‌​‌.American Institute for​​ Aeronautics and Astronautics1998​​​‌back to text
• 52‌ inproceedingsG.Guillaume Couégnat‌​‌, G. L.Gérard​​​‌ L. Vignoles, V.​Virginie Drean, C.​‌Christianne Mulat, W.​​William Ros, G.​​​‌Grégory Perrot, T.​Thomas Haurat, J.​‌Jalal El-Yagoubi, M.​​Martin Eric, M.​​​‌Mario Ricchiuto, C.​Christian Germain and M.​‌Michel Cataldi. Virtual​​ material approach to self-healing​​​‌ CMCs.4th European​ Conference for Aerospace Sciences​‌ (EUCASS)Saint Petersurg, Russia​​July 2011HALback​​​‌ to text
• 53 book​C. M.Constantine M.​‌ Dafermos. Hyperbolic conservation​​ laws in continuum physics​​​‌.325Grundlehren der​ Mathematischen WissenschaftenBerlin: Springer​‌2016back to text​​
• 54 inproceedingsZ.Z.​​​‌ Demirbilek, A.A.​ Zundel and O.O.​‌ Nwogu. Boussinesq Modeling​​ of Wave Propagation and​​​‌ Runup over Fringing Coral​ Reefs, Model Evaluation Report​‌.Coastal and Hydraulics​​ Laboratory Technical Note CHLTR0712​​​‌Vicksburg, MS: U.S. Army​ Engineer Research and Development​‌ Center2007back to​​ text
• 55 articleC.​​​‌Cécile Dobrzynski, P.​Pascal Frey, B.​‌Bijan Mohammadi and O.​​Olivier Pironneau. Fast​​​‌ and accurate simulations of​ air-cooled structures.Computer​‌ Methods in Applied Mechanics​​ and Engineering1952006​​​‌, 3168-3180HALback​ to text
• 56 inproceedings​‌C.Claes Eskilsson,​​ J.Johannes Palm,​​​‌ A. P.Allan Peter​ Engsig-Karup, U.Umberto​‌ Bosi and M.Mario​​ Ricchiuto. Wave Induced​​​‌ Motions of Point-Absorbers: a​ Hierarchical Investigation of Hydrodynamic​‌ Models.11th European​​ Wave and Tidal Energy​​​‌ Conference (EWTEC)Nantes, France​September 2015HALback​‌ to textback to​​ text
• 57 articleA.​​​‌A.G Filippini, M.​M. Kazolea and M.​‌M. Ricchiuto. A​​ flexible genuinely nonlinear approach​​​‌ for nonlinear wave propagation,​ breaking and runup.​‌Journal of Computational Physics​​3102016back to​​​‌ text
• 58 articleH.​Hans Forrer and R.​‌Rolf Jeltsch. A​​ Higher-Order Boundary Treatment for​​​‌ Cartesian-Grid Methods.Journal​ of Computational Physics140​‌21998, 259-277​​URL: https://www.sciencedirect.com/science/article/pii/S0021999198958910DOIback​​​‌ to text
• 59 article​E.Elena Gaburro,​‌ M. J.Manuel J​​ Castro and M.Michael​​​‌ Dumbser. A well​ balanced finite volume scheme​‌ for general relativity.​​SIAM Journal on Scientific​​​‌ Computing4362021​, B1226--B1251back to​‌ text
• 60 articleE.​​Elena Gaburro, M.​​​‌ J.Manuel J Castro​ and M.Michael Dumbser​‌. Well-balanced Arbitrary-Lagrangian-Eulerian finite​​ volume schemes on moving​​​‌ nonconforming meshes for the​ Euler equations of gas​‌ dynamics with gravity.​​Monthly Notices of the​​​‌ Royal Astronomical Society477​22018, 2251--2275​‌back to text
• 61​​ articleE.Edwige Godlewski​​​‌, M.Martin Parisot​, J.Jacques Sainte-Marie​‌ and F.Fabien Wahl​​. Congested shallow water​​​‌ model: on floating body​.The SMAI journal​‌ of computational mathematics6​​2020, 227--251URL:​​​‌ https://smai-jcm.centre-mersenne.org/item/SMAI-JCM_2020__6__227_0/DOIback to​ textback to text​‌
• 62 articleE.Edwige​​ Godlewski, M.Martin​​​‌ Parisot, J.Jacques​ Sainte-Marie and F.Fabien​‌ Wahl. Congested shallow​​ water model: roof modeling​​​‌ in free surface flow​.ESAIM: Mathematical Modelling​‌ and Numerical Analysis52​​5Sep 2018,​​ 1679--1707URL: http://dx.doi.org/10.1051/m2an/2018032DOI​​​‌back to text
• 63‌ inproceedingsH.H. Hebert‌​‌, S.S. Abadie​​, M.M. Benoit​​​‌, R.R. Creach‌, A.A. Frere‌​‌, A.A. Gailler​​, S.S. Garziglia​​​‌, Y.Y. Hayashi‌, A.A. Lovenbruck‌​‌, O.O. Macary​​, R.R. Marcer​​​‌, D.D. Morichon‌, R.R. Pedreros‌​‌, C.C. Rebour​​, M.M. Ricchiuto​​​‌, F.F. Schinderle‌, R. S.R.‌​‌ Silva Jacinto, M.​​M. Terrier, S.​​​‌S. Toucanne, P.‌P. Traversa and D.‌​‌D. Violeau. Project​​ TANDEM (Tsunamis in the​​​‌ Atlantic and the English‌ ChaNnel: Definition of the‌​‌ Effects through numerical Modeling)​​ (2014-2018): a French initiative​​​‌ to draw lessons from‌ the Tohoku-oki tsunami on‌​‌ French coastal nuclear facilities​​.Geophysical Research Abstracts,​​​‌ EGU2014-6421-1, Volume 16EGU‌ General Assembly 20142014‌​‌back to text
• 64​​ inproceedingsE.E. Josyula​​​‌ and J.J. Burt‌. Review of Rarefied‌​‌ gas effects in hypersonic​​ applications.RTO AVT/VKI​​​‌ Lecture Series - Models‌ and Computational Methods for‌​‌ Rarefied Flows2011back​​ to text
• 65 article​​​‌M.M. Kazolea,‌ A.A.I. Delis and‌​‌ C.C.E. Synolakis.​​ Numerical treatment of wave-breaking​​​‌ on unstructured finite volume‌ approximations for extended Boussinesq-type‌​‌ equations.Journal of​​ Computational Physics2712014​​​‌back to text
• 66‌ articleA.A.B. Kennedy‌​‌, J. ..J​​ .T. Kirby and R.​​​‌R.A. Dalrymple. Boussinesq‌ modeling of wave transformation,‌​‌ breaking and runup. Part​​ I: 1D.J.​​​‌ Waterw. PortCoast. Ocean Eng.‌1262000, 39--47‌​‌back to text
• 67​​ inproceedingsN.N. Kroll​​​‌, T.T. Leicht‌, C.Ch. Hirsch‌​‌, F.F. Bassi​​, C.C. Johnston​​​‌, K.K.A. Sorensen‌ and K.K. Hillewaert‌​‌. Results and Conclusions​​ of the European Project​​​‌ IDIHOM on High-Order Methods‌ for Industrial Aerodynamic Applications‌​‌.53rd AIAA Aerospace​​ Sciences Meeting2015back​​​‌ to text
• 68 article‌M.M. Lesoinne and‌​‌ C.C. Farhat.​​ Geometric Conservation Laws for​​​‌ flow problems with moving‌ boundaries and deformable meshes,‌​‌ and their impact on​​ aeroelastic computations.Comput.​​​‌ Method Appl. M.134‌1-21996, 71--90‌​‌back to text
• 69​​ articleA.A. Loseille​​​‌ and F.F. Alauzet‌. Continuous mesh framework,‌​‌ Part II: validations and​​ applications.SIAM in​​​‌ Numerical Analysis491‌2011back to text‌​‌
• 70 inproceedingsA.A.​​ Loseille, A.A.​​​‌ Dervieux, P.P.J.‌ Frey and F.F‌​‌ Alauzet. Achievement of​​ global second-order mesh convergence​​​‌ for discontinuous flows with‌ adapted unstructured meshes.‌​‌18th AIAA Computational Fluid​​ Dynamics Conference, Miami (FL)​​​‌AIAA paper 2007-41862007‌back to text
• 71‌​‌ articleK.Ken Mattsson​​ and M.Martin Almquist​​​‌. A high-order accurate‌ embedded boundary method for‌​‌ first order hyperbolic equations​​.Journal of Computational​​​‌ Physics3342017,‌ 255-279URL: https://www.sciencedirect.com/science/article/pii/S0021999116306969DOI‌​‌back to text
• 72​​ articleL.L. Nouveau​​​‌, H.H. Beaugendre‌, C.C. Dobrzynski‌​‌, R.R. Abgrall​​​‌ and M.M. Ricchiuto​. An explicit, adaptive,​‌ residual based splitting approach​​ for the steardy and​​​‌ time dependent penalized Navier​ Stokes equations.Comput.​‌ Meth. Appl. Mech. Engrg.​​3032016, 208--230​​​‌back to textback​ to text
• 73 inproceedings​‌O.O. Nwogu.​​ Numerical prediction of breaking​​​‌ waves and currents with​ a Boussinesq model.​‌Proceedings 25th International Conference​​ on Coastal Engineering1996​​​‌back to text
• 74​ inproceedingsM.M. Papadakis​‌, H.H.W. Yeong​​ and I.I.G. Suares​​​‌. Simulation of Ice​ Shedding from a Business​‌ Jet Aircraft.AIAA,​​ Aerospace Sciences Meeting and​​​‌ Exhibit, 45 th, Reno,​ NVAIAA Paper 2007-506​‌2007back to text​​
• 75 articleM.M.​​​‌ Peng, Z.Z.​ Sun and K.K.​‌ Wu. OEDG: Oscillation-eliminating​​ discontinuous Galerkin method for​​​‌ hyperbolic conservation laws.​Math.Comp.942025,​‌ 1147-1198back to text​​
• 76 articleM.M.​​​‌ Ricchiuto. On the​ C-property and Generalized C-property​‌ of Residual Distribution for​​ the Shallow Water Equations​​​‌.Journal of Scientific​ Computing4812011​‌, 304-318back to​​ text
• 77 bookP.​​​‌ J.P J Roache​. Verification and validation​‌ in computational science and​​ engineering.Albuquerque, NM​​​‌Hermosa1998, URL:​ https://cds.cern.ch/record/580994back to text​‌
• 78 articleV.V.​​ Roeber, K.K.F.​​​‌ Cheung and M.M.H.​ Kobayashi. Shock-capturing Boussinesq-type​‌ model for nearshore wave​​ processes.Coastal Engineering​​​‌572010, 407--423​back to text
• 79​‌ inproceedingsJ.J. Slotnick​​, A.A. Khodadoust​​​‌, J.J. Alonso​, D.D. Darmofal​‌, W.W. Gropp​​, E.E. Lurie​​​‌ and D.D. Mavriplis​. CFD Vision 2030​‌ Study: A Path to​​ Revolutionary Computational Aerosciences.​​​‌NASA/CR-2014-2181782014back to​ text
• 80 articleS.​‌Sirui Tan and C.-W.​​Chi-Wang Shu. Inverse​​​‌ Lax-Wendroff procedure for numerical​ boundary conditions of conservation​‌ laws.Journal of​​ Computational Physics22921​​​‌2010, 8144-8166URL:​ https://www.sciencedirect.com/science/article/pii/S0021999110003979DOIback to​‌ text
• 81 articleP.​​ D.P. D. Thomas​​​‌ and C. K.C.​ K. Lombard. Geometric​‌ Conservation Law and Its​​ Application to Flow Computations​​​‌ on Moving Grids.​AIAA Journal1710​‌1979, 1030-1037back​​ to text
• 82 article​​​‌M.M. Tissier,​ P.P. Bonneton,​‌ F.F. Marche,​​ F.F. Chazel and​​​‌ D.D. Lannes.​ A new approach to​‌ handle wave breaking in​​ fully non-linear Boussinesq models​​​‌.Coastal Engineering67​2012, 54--66URL:​‌ http://www.sciencedirect.com/science/article/pii/S0378383912000749DOIback to​​ text
• 83 articleM.​​​‌M. Tonelli and i.​i M. Pett.​‌ Simulation of wave breaking​​ over complex bathymetries by​​​‌ a Boussinesq model.​Journal of Hydraulic Research​‌4942011,​​ 473-486back to text​​​‌
• 84 articleM.M.​ Tonelli and M.M.​‌ Petti. Hybrid finite​​ volume-finite difference scheme for​​​‌ 2DH improved Boussinesq equations​.Coastal Engineering 56​‌2009, 609--620back​​ to text
• 85 inproceedings​​​‌P.P. Tran,​ P.P. Benquet,​‌ G.G. Baruzzi and​​ W.W.G. Habashi.​​ Design of Ice Protection​​​‌ Systems and Icing Certification‌ Through Cost - Effective‌​‌ Use of CFD.​​AIAA Aerospace Sciences Meeting​​​‌ and ExhibitAIAA-CP 2002-0382‌2002back to text‌​‌
• 86 articleM.M.D.​​ de Tullio, P.​​​‌ D.P. De Palma‌, G.G. Iaccarino‌​‌, G.G. Pascazio​​ and M.M. Napolitano​​​‌. An immersed boundary‌ method for compressible flows‌​‌ using local grid refinement​​.Journal of Computational​​​‌ Physics22522007‌, 2098-2117URL: https://www.sciencedirect.com/science/article/pii/S0021999107001222‌​‌DOIback to text​​
• 87 articleZ.Z.J.​​​‌ Wang, K.K.‌ Fidkowski, R.R.‌​‌ Abgrall, F.F.​​ Bassi, D.D.​​​‌ Caraeni, A.A.‌ Cary, H.H.‌​‌ Deconinck, R.R.​​ Hartmann, K.K.​​​‌ Hillewaert, H.H.T.‌ Huynh, N.N.‌​‌ Kroll, G.G.​​ May, P.-O.P.-O.​​​‌ Persson, B.B.‌ van Leer and M.‌​‌M. Visbal. High-order​​ CFD methods: current status​​​‌ and perspective.International‌ Journal for Numerical Methods‌​‌ in Fluids728​​2013, 811--845back​​​‌ to text
• 88 inproceedings‌J.J. Weber.‌​‌ WEC Technology Readiness and​​ Performance Matrix, finding the​​​‌ best research technology development‌ trajectory.4th International‌​‌ Conference on Ocean Engineering​​ - ICOE2012back​​​‌ to textback to‌ text