2.1 Overall objectives

Together with the diffusion of scientific computing, there has been a recent and impressive increase of the demand for numerical methods. The problems to be addressed are everyday more complex and require specific numerical algorithms. The quality of the results has to be accurately assessed, so that in-silico experiments results can be trusted. Nowadays, producing such reliable numerical results goes way beyond the abilities of isolated researchers, and must be carried out by structured teams.

The topics addressed by the RAPSODI project-team belong to the broad theme of numerical methods for the approximation of solutions of systems of partial differential equations (PDEs). Besides standard convergence properties, a good numerical method for approximating a physical problem has to satisfy at least the following three criteria:

1. preservation at the discrete level of some crucial features of the solution, such as positivity of solutions, conservation of prescribed quantities (e.g., mass), the decay of physically motivated entropies, etc;
2. provide accurate numerical approximations at a reasonable computational cost (and ultimately maximize the accuracy at a fixed computational effort);
3. robustness with respect to physical conditions: the computational cost for a given accuracy should be essentially insensitive to a change of physical parameters.

We aim to develop methods fulfilling the above quality criteria for physical models which all display a dissipative behavior, and that are motivated by industrial collaborations or multidisciplinary projects. In particular, we have identified a couple of specific situations we plan to investigate: models from corrosion science (in the framework of nuclear waste repository) 65, low-frequency electromagnetism 84, and mechanics of complex inhomogeneous fluids arising in avalanches 75 or in porous media 67.

Ideally, we should allow ourselves to design entirely new numerical methods. For some applications however (often in the context of industrial collaborations), the members of the team have to compose with existing codes. The numerical algorithms have thus to be optimized under this constraint.

2.2 Scientific context

Some technological bottlenecks related to points (a)–(c) mentioned above are well identified. In particular, it appears that a good numerical method should handle general meshes, so that dynamic mesh adaptation strategies can be used in order to achieve (b). But it should also be of the highest possible order while remaining stable in the sense of (a), and robust in the sense of (c). There have been numerous research contributions on each point of (a)–(c) in the last decades, in particular for solving each difficulty separately, but combining them still leads to unsolved problems of crucial interest.

Let us mention for example the review paper by Jérôme Droniou 89, where it is highlighted that all the linear methods for solving diffusion equations on general meshes suffer from the same lack of monotonicity and preserve neither the positivity of the solutions nor the decay of the entropy. Moreover, there is no complete convergence proof for the nonlinear methods exposed in 89. The first convergence proof for a positivity preserving and entropy diminishing method designed to approximate transient dissipative equations on general meshes was proposed recently in 76. The idea and the techniques introduced in 76 should be extended to practical applications.

In systems of PDEs, the values of the physical parameters often change the qualitative behavior of the solutions. Then, one challenge in the numerical approximation of such systems is the design of methods which can be applied for a large range of parameters, as in particular in the regime of singular perturbations. Such schemes, called asymptotic-preserving (AP) schemes 96, are powerful tools as they allow the use of the same scheme for a given problem and for its limit with fixed discretization parameters. In many cases, the AP property of numerical schemes is just empirically established, without any rigorous proof. We aim to extend the techniques recently introduced in 72 for the drift-diffusion system, and based on the control of the numerical dissipation of entropy, to other dissipative systems in order to prove the AP property of numerical schemes.

The question of the robustness of the numerical methods with respect to the physical parameters is also fundamental for fluid mixture models. The team already developed such schemes for the variable density Navier–Stokes system 74, 75. We aim to propose new ones for more complex models with the same philosophy in mind. On the one hand, we will be interested in high-order schemes, which must be as simple as possible in view of 3D practical implementations. Let us stress that combining high order accuracy and stability is very challenging. On the other hand, the optimization of the computations will have to be considered, in particular with the development of some a posteriori error estimators. Impressive progresses have been achieved in this field 87, allowing important computational savings without compromising the accuracy of the results. Recently, we successfully applied this strategy to the Reissner–Mindlin model arising in solid mechanics 86, the dead-oil model for porous media flows 78, or the Maxwell equations in their quasi-static approximation for some eddy current problems 84, 85. We aim to develop new a posteriori estimators for other dissipative systems, like fluid mixture models.

In a nutshell, our goal is to take advantage of and extend the most recent breakthroughs of the mathematical community to tackle in an efficient way some application-guided problems coming either from academics or from industrial partners. To this end, we shall focus on the following objectives, which are necessary for the applications we have in mind:

1. Design and analysis of structure-preserving numerical methods.
2. Computational optimization.

3.1 Design and analysis of structure-preserving schemes

3.2 Optimizing the computational efficiency

4.1 Porous media flows

Porous media flows are of great interest in many contexts, like, e.g., oil engineering, water resources management, nuclear waste repository management, or carbon dioxide sequestration. We refer to 67, 66 for an extensive discussion on porous media flow models.

From a mathematical point of view, the transport of complex fluids in porous media often leads to possibly degenerate parabolic conservation laws. The porous rocks can be highly heterogeneous and anisotropic. Moreover, the grids on which one intends to solve numerically the problems are prescribed by the geological data, and might be nonconformal with cells of various shapes. Therefore, the schemes used for simulating such complex flows must be particularly robust.

4.2 Corrosion and concrete carbonation

The team is interested in the theoretical and numerical analysis of mathematical models describing the degradation of materials, as concrete carbonation and corrosion. The study of such models is an important environmental and industrial issue. Atmospheric carbonation degrades reinforced concretes and limits the lifetime of civil engineering structures. Corrosion phenomena issues occur for instance in the reliability of nuclear power plants and the nuclear waste repository. The study of the long time evolution of these phenomena is of course fundamental in order to predict the lifetime of the structures.

From a mathematical point of view, the modeling of concrete carbonation (see 61) as the modeling of corrosion in an underground repository (DPCM model developed by Bataillon et al. 65) lead to systems of PDEs posed on moving domains. The coupling between convection-diffusion-reaction equations and moving boundary equations leads to challenging mathematical questions.

4.3 Complex fluid flows

The team is interested in numerical methods for the simulation of systems of PDEs describing complex flows, like for instance mixture flows, granular gases, rarefied gases, or quantum fluids.

Variable-density, low-Mach flows have been widely studied in the recent literature because of their applicability in various phenomena such as flows in high-temperature gas reactors, meteorological flows, flows with convective and/or conductive heat transfer or combustion processes. In such cases, the resolution of the full compressible Navier–Stokes system is not adapted, because of the sound waves' speed. The Boussinesq incompressible model is not a better alternative for such low-speed phenomena, because the compressibility effects cannot be totally cancelled due to large variations of temperature and density. Consequently, some models have been formally derived, leading to the filtering of the acoustic waves by the use of some formal asymptotic expansions and two families of methods have been developed in the literature in order to compute these flows. We are interested in particular in the so-called pressure-based methods, which are more robust than density-based solvers, although their range of validity is in general more limited.

The kinetic theory of molecular gases models a gas as a system of elastically colliding spheres, conserving mechanical energy during impact. Once initialized, it takes to a molecular gas no more than a few collisions per particle to relax to its equilibrium state, characterized by a Maxwellian velocity distribution and a certain homogeneous density (in the absence of external forces). A granular gas is a system of dissipatively colliding, macroscopic particles (grains). This slight change in the microscopic dynamics (converting energy into heat) causes drastic changes in the behavior of the gas: granular gases are open systems, which exhibit self-organized spatio-temporal cluster formations, and have no equilibrium distribution. They can be used to model silos, avalanches, pollen or planetary rings.

Quantum models can be used to describe superfluids, quantum semiconductors, weakly interacting Bose gases, or quantum trajectories of Bohmian mechanics. They have attracted considerable attention in the last decades, due in particular to the development of nanotechnology applications. To describe quantum phenomena, there exists a large variety of models. In particular, there exist three different levels of description: microscopic, mesoscopic, and macroscopic. The quantum Navier–Stokes equations deal with a macroscopic description in which the quantum effects are taken into account through a third-order term called the quantum Bohm potential. This Bohm potential arises from the fluid dynamical formulation of the single-state Schrödinger equation. The nonlocality of quantum mechanics is approximated by the fact that the equations of state do not only depend on the particle density but also on its gradient. These equations were employed to model field emissions from metals and steady-state tunneling in metal-insulator-metal structures, and to simulate ultra-small semiconductor devices.

4.4 Stratigraphy

The knowledge of the geology is a prerequisite before simulating flows within the subsoil. Numerical simulations of the geological history thanks to stratigraphy numerical codes allow to complete the knowledge of the geology where experimental data are lacking. Stratigraphic models consist in a description of the erosion and sedimentation phenomena at geological scales.

The characteristic time scales for the sediments are much larger than the characteristic time scales for the water in the river. However, the (time-averaged) water flux plays a crucial role in the evolution of the stratigraphy. Therefore, defining appropriate models that take the coupling between the rivers and the sediments into account is fundamental and challenging. Once the models are at hand, efficient numerical methods must be developed.

4.5 Low-frequency electromagnetism

Numerical simulation is nowadays an essential tool in order to design electromagnetic systems, by estimating the electromagnetic fields generated in a wide variety of devices. An important challenge for many applications is to quantify the intensity of the electric field induced in a conductor by a current generated in its neighborhood. In the low-frequency regime, we can for example cite the study of the impact on the human body of a high-tension line or, for higher frequencies, the one of a smartphone. But the ability to simulate accurately some electromagnetic fields is also very useful for nondestructive control, in the context of the maintenance of nuclear power stations for example. The development of efficient numerical tools, among which a posteriori error estimators, is consequently necessary to reach a high precision of calculation in order to provide estimations that are as reliable as possible.

5.1 Promotions

C. Cancès was promoted senior Inria researcher (DR2) as of October 2021.

I. Lacroix-Violet was promoted from associate professor at Université de Lille to full professor at Université de Lorraine (Nancy, France). She left the project-team at the end of August 2021.

5.2 Award

On November 23, 2021 C. Cancès was awarded the Blaise Pascal Prize from the Académie des Sciences: see here. The Blaise Pascal Prize is awarded every year by the Académie des Sciences after consultation of the SMAI-GAMNI group. It aims at rewarding outstanding achievements on the devising and mathematical analysis of numerical methods for the solution of PDEs realized by an under-40 researcher in France.

5.3 ABPDE IV conference

From November 16 to November 19, 2021 was held in Lille (at Polytech Lille engineering school) the 4th edition of the conference "Asymptotic Behaviors of systems of PDEs arising in Physics and Biology" (ABPDE IV), organized by C. Cancès, C. Chainais-Hillairet, M. Herda, I. Lacroix-Violet, Alexandre Mouton (Université de Lille), and T. Rey. This event gathered around 70 participants. The ABPDE conference has now become a well-established recurrent event (every 3 years); see also Section 10.1.1.

6.1 Platform ParaSkel++

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ParaSkel++ 54 is a C++ platform (freely distributed under LGPL), developed by L. Beaude and S. Lemaire, for the high-performance, arbitrary-order, 2/3D numerical approximation of PDEs on general polytopal meshes using skeletal Galerkin methods. Skeletal Galerkin methods are a vast family of numerical methods for the approximation of PDE-based models that satisfy the following two building principles (see 27):

• $•$
  the degrees of freedom (DOF) of the method split into (i) skeleton DOF, attached to the geometric entities (vertices, edges, faces) composing the mesh skeleton and common to all cells sharing the geometric entity in question, which prescribe the conformity properties of the underlying discrete functional space, and (ii) bulk DOF (if need be), attached to the interior of the cells, which play no role in the prescription of the conformity properties of the underlying discrete functional space;
• $•$
  the global discrete bilinear form of the problem (potentially after linearization, if the problem is nonlinear) writes as the sum over the mesh cells of cell-wise (referred to as local) bilinear contributions.

The very structure underpinning skeletal methods grants them the property of being amenable to static condensation, i.e., locally to each cell, bulk DOF can be eliminated in terms of the local skeleton DOF by means of a Schur complement. The final global system to solve thus writes in terms of the skeleton DOF only. The skeletal family encompasses in particular standard FE methods and virtual-like Galerkin methods (VEM, HHO, HDG...). It does not contain (plain vanilla) DG methods. ParaSkel++ offers a high-performance factorized C++ architecture for the implementation of arbitrary-order skeletal methods on general 2/3D polytopal meshes. A first version (v1, August 2021) of the platform is operational, featuring a sequential implementation of all the main skeletal methods. The next crucial development steps will be the parallelization on shared memory (before considering distributed memory), and the implementation of efficient quadrature formulas on polytopal cells. Eventually, the ParaSkel++ platform is expected to possess five main assets with respect to other codes of the same nature from the community: (i) a unified 2/3D implementation, (ii) the native support of any type of DOF (vertex-, edge-, face-, and cell-based), (iii) an ultra-factorized architecture (with common-to-all-methods local elimination and global assembly steps), (iv) the use of efficient quadrature formulas on general polytopal cells (without the need for subtessellation), and (v) the embedding of parallel computation facilities.

6.2 Code KINEBEC

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KINEBEC (Kinetic Bose–Einstein Condensates) 55 is a C code (freely distributed under GPL), developed by Alexandre Mouton (CNRS permanent engineer at Université de Lille) and T. Rey, devoted to the simulation of collisional kinetic equations of Boltzmann type using a deterministic, spectral Galerkin approach. While mainly devoted to the numerical simulation of the Boltzmann–Nordheim equation (BNE) for fermions and bosons, this code can also be used to solve the classical Boltzmann equation (BE). It relies on state-of-the-art fast spectral approaches to solve with high accuracy and efficiency both the BNE and BE. It has been parallelized on shared memory (OpenMP), but also has a native MPI support for heterogeneous architectures, as well as CUDA capabilities.

6.3 Other codes

7.1 Modeling and numerical simulation of complex fluids

In 34, C. Cancès et al. establish an error estimate, within the generic framework for the spatial discretisation of partial differential equations of the Gradient Discretisation Method (GDM), for a class of degenerate parabolic problems. This result is obtained under very mild regularity assumptions on the exact solution. Their study covers well-known models like the porous medium equation and the fast diffusion equations, as well as the strongly degenerate Stefan problem. Several schemes are then compared in a last section devoted to numerical results.

In 44, C. Cancès and his co-author prove the existence of weak solutions to a system of two diffusion equations that are coupled by a pointwise volume constraint. The time evolution is given by gradient dynamics for a free energy functional. Their primary example is a model for the demixing of polymers, the corresponding energy is the one of Flory, Huggins and de Gennes. Due to the nonlocality in the equations, the dynamics considered here is qualitatively different from the one found in the formally related Cahn–Hilliard equations. Their angle of attack stems from the theory of optimal mass transport, that is, they consider the evolution equations for the two components as two gradient flows in the Wasserstein distance with one joint energy functional that has the volume constraint built in. The main difference with their previous work 77 is the nonlinearity of the energy density in the gradient part, which becomes singular at the interface between pure and mixed phases.

In 12, S. Bassetto, C. Cancès et al. are concerned with the numerical approximation of the Richards equation in a heterogeneous domain, each subdomain of which is homogeneous and represents a rocktype. Their first contribution is to rigorously prove convergence toward a weak solution of cell-centered finite volume schemes with upstream mobility and without Kirchhoff’s transform. Their second contribution is to numerically demonstrate the relevance of locally refining the grid at the interface between subregions, where discontinuities occur, in order to preserve an acceptable accuracy for the results computed with the schemes under consideration.

In 42, S. Bassetto, C. Cancès et al. further benchmark several numerical approaches building on upstream mobility two-point flux approximation finite volumes to solve Richards' equation in domains made of several rocktypes. Their study encompasses four different schemes corresponding to different ways to approximate the nonlinear transmission condition systems arising at the interface between different rocks, as well as different resolution strategies based on Newton's method with variable switch. The different methods are compared on filling and drainage test-cases with standard nonlinearities of Brooks–Corey and van Genuchten type, as well as with challenging steep nonlinearities.

In 15, C. Chainais-Hillairet et al. present and apply a computer-assisted method in order to prove the existence of traveling wave solutions to the Diffusion Poisson Coupled Model arising in corrosion modeling. They also establish a precise and certified description of the solutions.

In 25, A. Natale et al. develop a novel particle discretization for compressible isentropic fluids and porous media flow. The main idea of the method is to replace the internal energy of the fluid by its Moreau–Yosida regularization in the $L^2$ sense, which can be efficiently computed as a semi-discrete optimal transport problem. Using a modulated energy argument which exploits the convexity of the energy in Eulerian variables, they prove quantitative convergence estimates towards smooth solutions.

In 35, T. Rey et al. review recent mathematical results in kinetic granular materials, especially for those which arose since the last review by Villani on the same subject. This model describes the nonequilibrium behavior of materials composed of a large number of interacting, nonnecessarily microscopic particles, such as grains or planetary rings. This theoretical knowledge is then used to validate a new high-order numerical method for this equation, highlighting through numerics some theoretical open problems.

7.2 Numerical simulation in low-frequency electromagnetism

In 24, E. Creusé et al. study the $𝐀-\phi -𝐁$ magnetodynamic Maxwell system, given in its potential and space-time formulation. First, the existence of strong solutions with the help of the theory of Showalter on degenerate parabolic problems is established. Second, using energy estimates, the existence and the uniqueness of the weak solution to the $𝐀-\phi -𝐁$ system is inferred.

In 23, S. Lemaire et al. prove a discrete version of the first Weber inequality on three-dimensional hybrid spaces spanned by vectors of polynomials attached to the elements and faces of a polyhedral mesh. They then introduce two Hybrid High-Order methods for the approximation of the magnetostatics model, in both its (first-order) field and (second-order) vector potential formulations. These methods are applicable on general polyhedral meshes, and allow for arbitrary orders of approximation. Leveraging the previously established discrete Weber inequality, they perform a comprehensive analysis of the two methods, that they finally validate on a set of test-cases.

7.3 Structure-preserving numerical methods

In 14, C. Chainais-Hillairet et al. develop their work 71. They establish a priori estimates which lead to the existence of a solution to the scheme, and they prove the exponential decay of the discrete relative entropy towards the thermal equilibrium. Moreover, numerical results assess the good behavior of the numerical schemes.

In 46, C. Chainais-Hillairet, M. Herda, S. Lemaire and J. Moatti devise and study three Hybrid Finite Volume methods for an heterogeneous and anisotropic linear advection-diffusion equation on general meshes. They consider two linear methods, as well as a new, nonlinear scheme. They prove the existence of a solution to each scheme, and positivity of the discrete solutions to the nonlinear scheme. For the three schemes, they show that the discrete solutions converge exponentially fast in time towards their associated discrete steady-states. Their theoretical results are illustrated by numerical simulations.

In 17, C. Cancès, C. Chainais-Hillairet, B. Gaudeul et al. consider an unipolar degenerate drift-diffusion system arising in the modeling of organic semiconductors. They design four different Finite Volume schemes based on four different formulations of the fluxes. They provide a stability analysis and existence results for the four schemes; the convergence is established for two of them.

In 49, B. Gaudeul and his co-author consider a nonlinear cross-diffusion system arising from the consideration of nonzero various ions sizes in a Nernst–Planck–Poisson model. For two different Finite Volume schemes based on two different formulations of the fluxes of the problem, they discuss stability and existence results. For both of them, they report a convergence proof under a non-degeneracy assumption. Numerical experiments illustrate the behavior of the schemes.

In 20, C. Cancès et al. propose a Finite Element scheme for the numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker–Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange Finite Elements of degree 1, it is locally conservative after a local post-processing giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.

In 19, C. Cancès and his co-author study a time-implicit Finite Volume scheme for the degenerate Cahn–Hilliard model proposed in 90 and studied mathematically in 77. The scheme is shown to preserve the key properties of the continuous model, namely mass conservation, positivity of the concentrations, the decay of the energy, and the control of the entropy dissipation rate. This allows to establish the existence of a solution to the nonlinear algebraic system corresponding to the scheme. Furthermore, thanks to compactness arguments, the approximate solution is shown to converge towards a weak solution to the continuous problem as the discretization parameters tend to 0. Numerical results illustrate the behavior of the numerical scheme.

In 43, C. Cancès et al. propose a provably convergent Finite Volume scheme for the so-called Stefan–Maxwell model, which describes the evolution of the composition of a multi-component mixture and reads as a cross-diffusion system. The proposed scheme relies on a Two-Point Flux Approximation, and preserves at the discrete level some fundamental theoretical properties of the continuous model, namely the non-negativity of the solutions, the conservation of mass, and the preservation of the volume-filling constraints. In addition, the scheme satisfies a discrete entropy-entropy dissipation relation, very close to the relation which holds at the continuous level. In this article, C. Cancès et al. present the scheme together with its numerical analysis, and finally illustrate its behavior with some numerical results.

In 45, C. Cancès and his co-auhor propose and study an implicit finite volume scheme for a general model which describes the evolution of the composition of a multi-component mixture in a bounded domain. They assume that the whole domain is occupied by the different phases of the mixture, which leads to a volume filling constraint. In the continuous model, this constraint yields the introduction of a pressure, which should be thought as a Lagrange multiplier for the volume filling constraint. The pressure solves an elliptic equation, to be coupled with parabolic equations, possibly including cross-diffusion terms, which govern the evolution of the mixture composition. The system admits an entropy structure, which is the cornerstone of the analysis. The main objective of their work is the design of a two-point flux approximation finite volume scheme which preserves the key properties of the continuous model, namely the volume filling constraint and the control of the entropy production. Thanks to these properties, and in particular to the discrete entropy-entropy dissipation relation, the authors are able to prove the existence of solutions to the scheme and its convergence. Finally, they illustrate the behavior of their scheme through different applications.

In 29, A. Natale and his co-author construct and analyze Two-Point Flux Approximation Finite Volume discretizations of the quadratic optimal transport problem in its dynamic form. They show numerically that these types of discretizations are prone to form instabilities in their more natural implementation, and propose a variation based on nested meshes in order to overcome these issues. Moreover, they introduce a strategy based on the barrier method to solve the discrete optimization problem.

In 48, E. Facca et al. give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge–Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerical solver based on the so-called dynamical Monge–Kantorovich approach, they propose a novel framework for the numerical approximation of the cut locus of a point in a manifold. They show the applicability of the proposed method on a few examples settled on 2d-surfaces embedded in ${ℝ}^3$ and discuss advantages and limitations.

In 30, T. Rey and his co-author focus on the stability properties of some recently introduced spectral methods that preserve equilibrium. Thanks to the high accuracy and the possibility to use fast algorithms, spectral methods represent an effective way to approximate the Boltzmann collision operator. On the other hand, the loss of some local invariants leads to the wrong long time behavior. A way to overcome this drawback, without sacrificing spectral accuracy, has been proposed recently with the construction of equilibrium-preserving spectral methods. Despite the ability to capture the steady-state with arbitrary accuracy, the theoretical properties of the method have never been studied in details. In this paper, using the perturbation argument developed by Filbet and Mouhot for the homogeneous Boltzmann equation, the authors prove stability, convergence, and spectrally accurate long-time behavior of the equilibrium-preserving approach.

In 52, T. Rey and his co-author introduce a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived directly using a constrained best approximation in the space of trigonometric polynomials and can be applied to a wide class of problems where preservation of moments is essential. The authors then apply the new spectral method to the evaluation of the Boltzmann collision term, and prove spectral consistency and stability of the resulting Fourier-Galerkin approximation scheme. They illustrate their theoretical findings by various numerical experiments.

In 41, R. Bailo and T. Rey propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and BGK equations. Projective integration has been recently proposed as a viable alternative to fully implicit and micro-macro methods for providing light, nonintrusive and almost AP integrators for collisional kinetic equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. The schemes are validated on a range of scenarios, demonstrating their prowess in dealing with extreme mass ratios, fluid instabilities, and other complex phenomena.

In 40, R. Bailo et al. develop finite volume schemes for the Cahn–Hilliard equation that unconditionally and discretely satisfy the boundedness of the phase field and the free-energy dissipation. Their numerical framework is applicable to a variety of free-energy potentials including the Ginzburg–Landau and Flory–Huggins, general wetting boundary conditions and degenerate mobilities. Its central thrust is the finite volume upwind methodology, which we combine with a semi-implicit formulation based on the classical convex-splitting approach for the free-energy terms. Extension to an arbitrary number of dimensions is straightforward thanks to their cost-saving dimensional-splitting nature, which allows to efficiently solve higher-dimensional simulations with a simple parallelization. The numerical schemes are validated and tested in a variety of prototypical configurations with different numbers of dimensions and a rich variety of contact angles between droplets and substrates.

In 39, R. Bailo et al. propose finite volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the particular case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever applicable. These results are illustrated through extensive numerical simulations which explore known behaviours in biology and showcase new phenomena not yet described by the literature.

In 38, M. Herda et al. introduce and analyze numerical schemes for the homogeneous and the kinetic Lévy–Fokker–Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of mass, heavy-tailed equilibrium and (hypo)coercivity properties. They perform a thorough analysis of the numerical schemes and show exponential stability. Along the way, they introduce new tools of discrete functional analysis, such as discrete nonlocal Poincaré and interpolation inequalities adapted to fractional diffusion. Their theoretical findings are illustrated and complemented with numerical simulations.

In 13, I. Lacroix-Violet et al. focus on the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analog of it. In particular, they give a rigorous proof of the order of the relaxation method (presented in 70 for cubic nonlinearities) and they propose a generalized version that allows to deal with general power law nonlinearities. Numerical simulations for different physical models show the efficiency of these methods.

7.4 Cost reduction for numerical methods

In 27, S. Lemaire presents a unifying viewpoint on Hybrid High-Order (HHO) 6 and Virtual Element (VE) 68 methods on general polytopal meshes in dimension 2 or 3, in terms of both formulation and analysis. The focus is on a model Poisson problem. To bridge the two paradigms, (i) he transcribes the (conforming) VE method into the HHO framework and (ii) proves $H^m$ approximation properties for the local polynomial projector in terms of which the local VE discrete bilinear form is defined. This allows him to perform a unified analysis of VE/HHO methods, that differs from standard VE analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to this unified analysis, he also studies interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases.

In 22, S. Lemaire et al. establish the equivalence between the Multiscale Hybrid-Mixed (MHM) 63 and the Multiscale Hybrid High-Order (MsHHO) 83 methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, they prove that the equivalence holds for general polytopal (coarse) meshes and arbitrary approximation orders. They finally leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods.

In 47, I. Lacroix-Violet and her co-author introduce a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the Runge–Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so.

In 51, T. Rey and his co-author present an efficient implementation of a spectral Fourier-Galerkin algorithm for the quantum Boltzmann–Nordheim equation (BNE) for fermions and bosons. The BNE was first formulated by Uehling and Uhlenbeck starting from a classical Boltzmann equation with heuristic arguments. Using novel parallelization techniques, they investigate some of the conjectured properties of the large time behavior of the solutions to this equation. In particular, they are able to observe numerically both Bose–Einstein condensation and Fermi–Dirac relaxation, and to make some conjectures on their stability.

In 18, C. Cancès and his co-author propose a reduced model for the migration of hydrocarbons in heterogeneous porous media. Their model keeps track of the time variable. This allows to compute steady-states that cannot be reached by the commonly used ray-tracing and invasion-percolation algorithms. An efficient Finite Volume scheme allowing for very large time steps is then proposed.

7.5 Asymptotic analysis

In 16, I. Lacroix-Violet et al. consider global weak solutions to compressible Navier–Stokes–Korteweg equations with density dependent viscosities, in a periodic domain $\Omega ={𝕋}^3$, with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-Sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition, introduced in 73. Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the density tends to infinity, I. Lacroix-Violet et al. conclude the exponential decay to equilibrium. The result also covers the quantum Navier–Stokes system with a drag term.

In 11, following the ideas of V. V. Zhikov and A. L. Pyatnitskii, and more precisely the stochastic two-scale convergence, B. Merlet et al. establish a homogenization theorem in a stochastic setting for two nonlinear equations: the equation of harmonic maps into the sphere, and the Landau–Lifshitz equation. Homogenization results for nonlinear problems are known to be difficult. In this particular case, the equations have strong nonlinear features; in particular, in general, their solutions are not unique. Here, the authors take advantage of the different equivalent definitions of weak solutions to the nonlinear problem to apply typical linear homogenization recipes.

In 53, F. Raimondi tackles the homogenization of a quasilinear elliptic problem having a singular lower-order term and posed in a two-component domain with an $ϵ$-periodic imperfect interface. A Dirichlet condition is prescribed on the exterior boundary, while the continuous heat flux is assumed to be proportional to the jump of the solution on the interface via a function of order ${ϵ}^{\gamma }$. An homogenization result for $-1<\gamma <1$ is proved by means of the periodic unfolding method, adapted to two-component domains by P. Donato, K. H. Le Nguyen and R. Tardieu. One of the main tools in the homogenization process is the study of a suitable auxiliary linear problem and a related convergence result. It shows that the gradient of $u^{ϵ}$ behaves like that of the solution of the auxiliary one, associated with a weak cluster point of the sequence $\left\{u^{ϵ}\right\}$, as $ϵ\to 0$. This allows not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate.

7.6 Applied calculus of variations

In 21, B. Merlet et al. establish new results on the approximation of $k$-dimensional surfaces ($k$-rectifiable currents) by polyhedral surfaces with convergence in $h$-mass and with preservation of the boundary (the approximating polyhedral surface has the same boundary as the limit). This approximation result is required in the convergence study of 82.

In 26, B. Merlet and his co-author study a family of functionals penalizing oblique oscillations. These functionals naturally appear in some variational problems related to pattern formation and are somewhat reminiscent of those introduced by Bourgain, Brezis and Mironescu to characterize Sobolev functions. More precisely, for a function $u$ defined on a tensor product ${\Omega }_1\times {\Omega }_2$, the family of functionals ${\left\{E_{\epsilon }\left(u\right)\right\}}_{\epsilon >0}$ that they consider vanishes if $u$ is of the form $u\left(x_1\right)$ or $u\left(x_2\right)$. They prove the converse property and related quantitative results. In particular, they describe the fine properties of functions with ${sup}_{\epsilon }{E}_{\epsilon }\left(u\right)<\infty$ by showing that, roughly, such $u$ is piecewise of the form $u\left(x_1\right)$ or $u\left(x_2\right)$ on domains separated by lines where the energy concentrates. It turns out that this problem naturally leads to the study of various differential inclusions, and has connections with branched transportation models.

In 31, M. Pegon studies large volume minimizers of isoperimetric problems derived from Gamow's liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. Considering a large class of potentials, given by general radial nonnegative kernels which are integrable on ${ℝ}^n$, such as Bessel potentials, M. Pegon proves the existence of minimizers of arbitrarily large mass, provided that the first moment of the kernels is below an explicit threshold. This contrasts with the case of Riesz potentials, where minimizers do not exist above a critical mass. In addition, renormalizing to a fixed volume, any sequence of minimizers converges to the ball as the mass goes to infinity. Finally, M. Pegon shows that the threshold on the first moment of the kernels is sharp, in the sense that large balls go from stable to unstable. A direct consequence of the instability of large balls above this threshold is that there exist nontrivial compactly supported kernels for which the problems admit minimizers which are not balls, that is, symmetry breaking occurs.

In 28, B. Merlet and M. Pegon consider an isoperimetric problem in which the standard perimeter $P\left(E\right)$ is replaced by $P\left(E\right)-\gamma P_{\epsilon }\left(E\right)$, with $0<\gamma <1$ and $P_{\epsilon }$ a nonlocal energy such that $P_{\epsilon }\left(E\right)\to P\left(E\right)$ as $\epsilon$ vanishes. They prove that unit area minimizers are disks for $\epsilon >0$ small enough. This isoperimetric problem is equivalent to a generalization of the liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial, sufficiently integrable kernel. In that formulation, their result states that if the first moment of the kernel is smaller than an explicit threshold, there exists a critical mass $m_0$ such that for any $m>m_0$, the disk is the unique minimizer of area $m$ up to translations.

In 50, motivated by some models of pattern formation involving an unoriented director field in the plane, B. Merlet, M. Pegon et al. study a family of unoriented counterparts to the Aviles–Giga functional. They introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which they call "trigonometric entropies". Using these tools they show two main theorems which parallel some results in the literature on the classical Aviles–Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. Their methods provide alternative proofs in the classical Aviles–Giga context.

8.1 Bilateral contracts with industry

The PhD thesis of S. Bassetto was funded by IFPEn. The contract followed the lines of the bilateral contract between Inria and IFPEn. S. Bassetto defended on December 16, 2021.

The PhD thesis of M. Jonval, that started in October 2021, is co-funded by Inria (salaries) and IFPEn (overhead costs). The contract follows the lines of the bilateral contract between Inria and IFPEn.

8.2 Bilateral grants with industry

CEA (Christian Bataillon) and ANDRA (Laurent Trenty) are involved in the EURAD project on corrosion modeling together with the RAPSODI project-team (C. Cancès, C. Chainais-Hillairet, B. Merlet, and F. Raimondi). More details on the project can be found in Section 9.2.1.

9.1 International research visitors

Between October 19 and November 19, C. Chainais-Hillairet and M. Herda invited Patricio Farrell, research group leader at WIAS (Berlin, Germany), and Dilara Abdel, PhD student within P. Farrell's group, to work on the numerical simulation of perovskite semiconductors. P. Farrell was invited as a visiting professor funded by the LabEx CEMPI (see Section 9.3.2), whereas D. Abdel received funding from the French Embassy in Berlin.

The team was also visited by national researchers.

S. Lemaire invited Théophile Chaumont-Frelet (Inria Nice) to visit him in Lille on November 8-10.

C. Cancès invited Flore Nabet (École Polytechnique) to visit him in Lille on June 15-17, as well as on December 5-6.

9.2 European initiatives

9.3 National initiatives

9.4 Regional initiatives

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

10.3 Popularization

11.1 Major publications

• 1 articleM.Marianne Bessemoulin-Chatard and C.Claire Chainais-Hillairet. Exponential decay of a finite volume scheme to the thermal equilibrium for drift--diffusion systems.Journal of Numerical Mathematics2532017, 147-168
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• 2 articleC.Caterina Calgaro, E.Emmanuel Creusé, T.Thierry Goudon and S.Stella Krell. Simulations of non homogeneous viscous flows with incompressibility constraints.Mathematics and Computers in Simulation1372017, 201-225
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• 3 articleC.Clément Cancès, T.Thomas Gallouët and L.Léonard Monsaingeon. Incompressible immiscible multiphase flows in porous media: a variational approach.Analysis & PDE1082017, 1845--1876
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• 4 articleC.Clément Cancès and C.Cindy Guichard. Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure.Foundations of Computational Mathematics1762017, 1525-1584
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• 5 inproceedingsC.Claire Chainais-Hillairet, B.Benoît Merlet and A.Alexis Vasseur. Positive Lower Bound for the Numerical Solution of a Convection-Diffusion Equation.FVCA8 2017 - International Conference on Finite Volumes for Complex Applications VIIILille, FranceSpringerJune 2017, 331-339
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• 6 articleD. A.Daniele Antonio Di Pietro, A.Alexandre Ern and S.Simon Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators.Computational Methods in Applied Mathematics144June 2014, 461-472
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• 7 articleG.Giacomo Dimarco, R.Raphaël Loubère, J.Jacek Narski and T.Thomas Rey. An efficient numerical method for solving the Boltzmann equation in multidimensions.Journal of Computational Physics3532018, 46-81
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• 8 articleF.Francis Filbet and M.Maxime Herda. A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure.Numerische Mathematik13732017, 535-577
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• 9 articleI.Ingrid Lacroix-Violet and A.Alexis Vasseur. Global weak solutions to the compressible quantum Navier--Stokes equation and its semi-classical limit.Journal de Mathématiques Pures et Appliquées1142018, 191-210
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• 10 articleB.Benôit Merlet. A highly anisotropic nonlinear elasticity model for vesicles I. Eulerian formulation, rigidity estimates and vanishing energy limit.Arch. Ration. Mech. Anal.21722015, 651--680
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11.2 Publications of the year

11.3 Other

11.4 Cited publications

• 56 articleR.R. Abgrall. A review of residual distribution schemes for hyperbolic and parabolic problems: the July 2010 state of the art.Commun. Comput. Phys.1142012, 1043--1080
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• 57 articleR.R. Abgrall, G.G. Baurin, A.A. Krust, D.D. de Santis and M.M. Ricchiuto. Numerical approximation of parabolic problems by residual distribution schemes.Internat. J. Numer. Methods Fluids7192013, 1191--1206
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• 58 articleR.R. Abgrall, A.A. Larat and M.M. Ricchiuto. Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes.J. Comput. Phys.230112011, 4103--4136
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• 59 articleR.R. Abgrall, A.A. Larat, M.M. Ricchiuto and C.C. Tavé. A simple construction of very high order non-oscillatory compact schemes on unstructured meshes.Comput. & Fluids3872009, 1314--1323
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• 60 techreportE.Etienne Ahusborde, B.Brahim Amaziane, A.Attila Baksay, G.Grażyna Bator, D.D. Becker, A.A. Bednar, M.M. Beres, R.R. Blaheta, Z.Z. Böhti, G.G. Bracke, L.L. Brazda, V.V. Brendler, K.Konstantin Brenner, J.J. Brezina, C.Clément Cancès, C.Claire Chainais-Hillairet, F.Florent Chave, F.Francis Claret, S.S. Domesova, V.Vaclava Havlova, M.M. Hokr, D.D. Horak, D.D. Jacques, F.F. Jankovsky, K.K. Kazymyrenko, O.O. Kolditz, T.T. Koudelka, T.T. Kovacs, T.T. Krejci, J.J. Kruis, E.E. Laloy, J.J. Landa, T.T. Lipping, D.D. Lukin, D.David Maš\'in, R.Roland Masson, J. C.J. C. L. Meeussen, M.M. Mollaali, A.A. Mon, L.L. Montenegro, V.Vanessa Montoya, G.Guillaume Pépin, J.J. Poonoosamy, N.N Prasianakis, Z.Z. Saâdi, J.J. Samper, G.Gianvito Scaringi, P.Pierre Sochala, C.Christophe Tournassat, K.K. Yoshioka and Y.Y. Yuankai. State Of the Art Report in the fields of numerical analysis and scientific computing. Final version as of 16/02/2020 deliverable D4.1 of the HORIZON 2020 project EURAD: European Joint Programme on Radioactive Waste Management.EURAD2020
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• 61 articleT.Toyohiko Aiki and A.Adrian Muntean. A free-boundary problem for concrete carbonation: front nucleation and rigorous justification of the $\sqrt{t}$-law of propagation.Interfaces Free Bound.1522013, 167--180
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• 62 articleB.B. Amaziane, A.A. Bergam, M.M. El Ossmani and Z.Z. Mghazli. A posteriori estimators for vertex centred finite volume discretization of a convection-diffusion-reaction equation arising in flow in porous media.Internat. J. Numer. Methods Fluids5932009, 259--284URL: http://dx.doi.org/10.1002/fld.1456
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• 63 articleR.R. Araya, C.C. Harder, D.D. Paredes and F.F. Valentin. Multiscale Hybrid-Mixed method.SIAM J. Numer. Anal.5162013, 3505--3531
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• 64 articleI.I. Babuška and W. C.W. C. Rheinboldt. Error estimates for adaptive finite element computations.SIAM J. Numer. Anal.1541978, 736--754
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• 65 articleC.C. Bataillon, F.F. Bouchon, C.C. Chainais-Hillairet, C.C. Desgranges, E.E. Hoarau, F.F. Martin, S.S. Perrin, M.M. Tupin and J.J. Talandier. Corrosion modelling of iron based alloy in nuclear waste repository.Electrochim. Acta55152010, 4451--4467
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• 66 bookJ.J. Bear and Y.Y. Bachmat. Introduction to modeling of transport phenomena in porous media.4Springer1990
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• 67 bookJ.J. Bear. Dynamic of Fluids in Porous Media.New YorkAmerican Elsevier1972
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• 68 articleL.L. Beirão da Veiga, F.F. Brezzi, A.A. Cangiani, G.G. Manzini, L.L.~D. Marini and A.A. Russo. Basic principles of virtual element methods.Math. Models Methods Appl. Sci. (M3AS)2312013, 199--214
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• 69 articleS.S. Berrone, V.V. Garbero and M.M. Marro. Numerical simulation of low-Reynolds number flows past rectangular cylinders based on adaptive finite element and finite volume methods.Comput. & Fluids402011, 92--112URL: http://dx.doi.org/10.1016/j.compfluid.2010.08.014
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• 70 phdthesisC.Christophe Besse. Analyse numérique des systèmes de Davey--Stewartson.Université Bordeaux 11998
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• 71 inproceedingsM.Marianne Bessemoulin-Chatard, C.Claire Chainais-Hillairet and H.Hélène Mathis. Numerical schemes for semiconductors energy- transport models.Finite Volumes for Complex Applications IXvol. 323 of Springer Proceedings in Mathematics and StatisticsFinite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples (R. Klöfkorn, E. Keilegavlen, F. A. Radu, and J. Fuhrmann, eds.),Bergen, NorwaySpringer, ChamJune 2020, pp. 75--90
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• 72 articleM.Marianne Bessemoulin-Chatard, C.Claire Chainais-Hillairet and M.-H.Marie-Hélène Vignal. Study of a fully implicit scheme for the drift-diffusion system. Asymptotic behavior in the quasi-neutral limit.SIAM, J. Numer. Anal.5242014
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• 73 miscD.D. Bresch, A.A. Vasseur and C.C. Yu. Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities.working paper or preprint2019, URL: https://arxiv.org/abs/1905.02701
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• 74 articleC.C. Calgaro, E.E. Creusé and T.T. Goudon. An hybrid finite volume-finite element method for variable density incompressible flows.J. Comput. Phys.22792008, 4671--4696
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• 75 articleC.Caterina Calgaro, E.Emmanuel Creusé and T.Thierry Goudon. Modeling and simulation of mixture flows: application to powder-snow avalanches.Comput. & Fluids1072015, 100--122URL: http://dx.doi.org/10.1016/j.compfluid.2014.10.008
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• 76 articleC.Clément Cancès and C.Cindy Guichard. Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equations.Mathematics of Computation852982016, 549-580
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• 77 articleC.C. Cancès, D.D. Matthes and F.F. Nabet. A two-phase two-fluxes degenerate Cahn--Hilliard model as constrained Wasserstein gradient flow.Arch. Ration. Mech. Anal.23322019, 837--866
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• 78 articleC.C. Cancès, I. S.I. S. Pop and M.M. Vohralík. An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow.Math. Comp.832852014, 153--188URL: http://dx.doi.org/10.1090/S0025-5718-2013-02723-8
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• 79 articleJ. A.J. A. Carrillo, A.A. Jüngel, P. A.P. A. Markowich, G.G. Toscani and A.A. Unterreiter. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities.Monatsh. Math.13312001, 1--82URL: http://dx.doi.org/10.1007/s006050170032
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• 80 inproceedingsC.C. Chainais-Hillairet. Entropy method and asymptotic behaviours of finite volume schemes.Finite volumes for complex applications. VII. Methods and theoretical aspects77Springer Proc. Math. Stat.Springer, Cham2014, 17--35
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• 81 articleC.C. Chainais-Hillairet, A.A. Jüngel and S.S. Schuchnigg. Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities.Modelisation Mathématique et Analyse Numérique5012016, 135-162
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• 82 articleA.A. Chambolle, L.L. Ferrari and B.B. Merlet. Variational approximation of size-mass energies for k-dimensional currents.ESAIM: Control, Optimisation and Calculus of Variations25 (2019)43September 2019, 39
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• 83 articleM.M. Cicuttin, A.A. Ern and S.S. Lemaire. A Hybrid High-Order method for highly oscillatory elliptic problems.Comput. Methods Appl. Math.1942019, 723--748
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• 84 articleE.E. Creusé, S.S. Nicaise, Z.Z. Tang, Y.Y. Le Menach, N.N. Nemitz and F.F. Piriou. Residual-based \it a posteriori estimators for the ${\bf A}-\varphi$ magnetodynamic harmonic formulation of the Maxwell system.Math. Models Methods Appl. Sci.2252012, 1150028-30URL: http://dx.doi.org/10.1142/S021820251150028X
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• 85 articleE.E. Creusé, S.S. Nicaise, Z.Z. Tang, Y.Y. Le Menach, N.N. Nemitz and F.F. Piriou. Residual-based a posteriori estimators for the $𝐓/$ magnetodynamic harmonic formulation of the Maxwell system.Int. J. Numer. Anal. Model.1022013, 411--429
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• 86 articleE.E. Creusé, S.S. Nicaise and E.E. Verhille. Robust equilibrated a posteriori error estimators for the Reissner-Mindlin system.Calcolo4842011, 307--335URL: http://dx.doi.org/10.1007/s10092-011-0042-0
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• 87 articleD. A.D. A. Di Pietro and M.M. Vohralík. A Review of Recent Advances in Discretization Methods, a Posteriori Error Analysis, and Adaptive Algorithms for Numerical Modeling in Geosciences.Oil & Gas Science and Technology-Rev. IFP(online first)June 2014, 1-29
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• 88 articleV.V. Dolejší, A.A. Ern and M.M. Vohralík. A framework for robust a posteriori error control in unsteady nonlinear advection-diffusion problems.SIAM J. Numer. Anal.5122013, 773--793URL: http://dx.doi.org/10.1137/110859282
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• 89 articleJ.J. Droniou. Finite volume schemes for diffusion equations: introduction to and review of modern methods.Math. Models Methods Appl. Sci.2482014, 1575-1620
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• 90 articleW.W. E and P.P. Palffy-Muhoray. Phase separation in incompressible systems.Phys. Rev. E554April 1997, R3844--R3846URL: https://link.aps.org/doi/10.1103/PhysRevE.55.R3844
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• 91 softwareV.V. Ehrlacher, C.C. Cancès and L.L. Monasse. Finite volume scheme for the Stefan--Maxwell model.1.02020,
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• 92 articleE.E. Emmrich. Two-step BDF time discretisation of nonlinear evolution problems governed by monotone operators with strongly continuous perturbations.Comput. Methods Appl. Math.912009, 37--62
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• 93 articleJ.J. de Frutos, B.B. García-Archilla and J.J. Novo. A posteriori error estimations for mixed finite-element approximations to the Navier-Stokes equations.J. Comput. Appl. Math.23662011, 1103--1122URL: http://dx.doi.org/10.1016/j.cam.2011.07.033
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• 94 articleL.Leslie Greengard and J.-Y.June-Yub Lee. Accelerating the nonuniform fast Fourier transform.SIAM Rev.4632004, 443--454URL: http://dx.doi.org/10.1137/S003614450343200X
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• 95 articleM. E.M. E. Hubbard and M.M. Ricchiuto. Discontinuous upwind residual distribution: a route to unconditional positivity and high order accuracy.Comput. & Fluids462011, 263--269URL: http://dx.doi.org/10.1016/j.compfluid.2010.12.023
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• 96 articleS.S. Jin. Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations.SIAM, J. Sci. Comput.211999, 441-454
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• 97 articleA.A. Majda and J.J. Sethian. The derivation and numerical solution of the equations for zero Mach number combustion.Combustion Science and Technology421985, 185--205
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• 98 softwareA.Alexandre Mouton, C.Caterina Calgaro and E.Emmanuel Creusé. NS2DDV - Navier-Stokes 2D à Densité Variable.April 2020
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• 99 articleF.F. Otto. The geometry of dissipative evolution equations: the porous medium equation.Comm. Partial Differential Equations261-22001, 101--174
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• 100 articleM.M. Ricchiuto and R.R. Abgrall. Explicit Runge-Kutta residual distribution schemes for time dependent problems: second order case.J. Comput. Phys.229162010, 5653--5691URL: http://dx.doi.org/10.1016/j.jcp.2010.04.002
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• 101 articleM.M. Vohralík. Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods.Numer. Math.11112008, 121--158URL: http://dx.doi.org/10.1007/s00211-008-0168-4
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