We study a two-point flux approximation finite volume scheme for a cross-diffusion system. The scheme is shown to preserve the key properties of the continuous systems, among which the decay of the entropy. The convergence of the scheme is established thanks to compactness properties based on the discrete entropy - entropy dissipation estimate. Numerical results illustrate the behavior of our scheme.
Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure
Inria Lille – Nord Europe（法国国家信息与自动化研究所里尔-北欧中心）；Sorbonne Universités, UPMC Univ. Paris 06, CNRS, INRIA, CEREMA, UMR 7598, Laboratoire Jacques-Louis Lions（索邦大学，巴黎第六大学，法国国家科学研究中心，法国国家信息与自动化研究所，法国城市与地区环境局，联合研究实验室7598，雅克-路易·利翁斯实验室）
A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the mass. The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved. The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suitable a priori estimates. The novelty is the extension of this inequality to the system…
An implicit Euler finite-volume scheme for an $n$-species population cross-diffusion system of Shigesada--Kawasaki--Teramoto-type in a bounded domain with no-flux boundary conditions is proposed and analyzed. The scheme preserves the formal gradient-flow or entropy structure and preserves the nonnegativity of the population densities. The key idea is to consider a suitable mean of the mobilities in such a way that a discrete chain rule is fulfilled and a discrete analog of the entropy inequality holds. The existence of finite-volume solutions, the convergence of the scheme, and the large-time …
In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure--valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong conver…
An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the one-dimensional torus, arising in population dynamics, is proposed and analyzed. The kernels are assumed to be in detailed balance and satisfy a weak cross-diffusion condition. The latter condition allows for negative off-diagonal coefficients and for kernels defined by an indicator function. The scheme preserves the nonnegativity of the densities, conservation of mass, and production of the Boltzmann and Rao entropies. The key idea is to ``translate'' the entropy calculations for the continuous equations to th…
In this paper, we develop a discretization for the non-linear coupled model of classical Darcy-Forchheimer flow in deformable porous media, an extension of the quasi-static Biot equations. The continuous model exhibits a generalized gradient flow structure, identifying the dissipative character of the physical system. The considered mixed finite element discretization is compatible with this structure, which gives access to a simple proof for the existence, uniqueness, and stability of discrete approximations. Moreover, still within the framework, the discretization allows for the development …
A class of high-order shock-capturing schemes, P$_n$T$_m$-BVD (Deng et al., J. Comp. Phys., 386:323-349, 2019; Comput. & Fluids, 200:104433, 2020.) schemes, have been devised to solve the Euler equations with substantially reduced numerical dissipation, which enable high-resolution simulations to resolve flow structures of wider range scales. In such simulations with low dissipation, errors of round-off level might grow and contaminate the numerical solutions. A typical example of such problems is the loss of symmetry in the numerical solutions for physical problems of symmetric configurations…
We consider an alternative Navier-Stokes model for compressible viscous ideal gases, originally proposed in \cite{Svard18}. We derive a priori estimates that are sufficiently strong to support a weak entropy solution of the system. Guided by these estimates, we propose a finite volume scheme, derive the analogous estimates and demonstrate grid convergence towards a weak entropy solution. Furthermore, this existence proof is valid for "large" initial data and no a priori assumptions on the solution are needed.
Fixed-point iterative sweeping methods were developed in the literature to efficiently solve steady state solutions of Hamilton-Jacobi equations and hyperbolic conservation laws. Similar as other fast sweeping schemes, the key components of this class of methods are the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. Furthermore, good properties of fixed-point iterative sweeping methods include that they have explicit forms and do not involve inverse operation of nonlinear local systems, and they can be applied to general hyperbolic equations using a…
In this paper, we propose a discretization of the multi-dimensional stationary compressible Navier-Stokes equations combining finite element and finite volume techniques. As the mesh size tends to 0, the numerical solutions are shown to converge (up to a subsequence) towards a weak solution of the continuous problem for ideal gas pressure laws p($ρ$) = a$ρ$ $γ$ , with $γ$ > 3/2 in the three-dimensional case. It is the first convergence result for a numerical method with adiabatic exponents $γ$ less than 3 when the space dimension is three. The present convergence result can be seen as a discre…
A Convergent Staggered Scheme for the Variable Density Incompressible Navier-Stokes Equations
Institut de Radioprotection et de Sûreté Nucléaire (IRSN)（法国核辐射防护与核安全研究院）；Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan（里昂大学，法国国家科学研究中心联合研究单位5208，里昂第一大学，卡米耶·若尔当研究所）
This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries in the finite volume approach. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. The new finite volume scheme has superior dispersion and dissipation properties in comparison to the compact reconstruction approach. These implicit gradients are re-used in viscous flux computation and post-processing, which further improves efficiency. A problem-independent shock…
The aim of this work is to 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 scheme proposed here relies on a two-point flux approximation, and preserves at the discrete level some fundamental theoretical properties of the continuous models, 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 dissipatio…
In this paper, we construct a novel Eulerian-Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine-Hugoniot junmp condition. Unfortunately, to avoid the …
For capillary driven flow the interface curvature is essential in the modelling of surface tension via the imposition of the Young--Laplace jump condition. We show that traditional geometric volume of fluid (VOF) methods, that are based on a piecewise linear approximation of the interface, do not lead to an interface curvature which is convergent under mesh refinement in time-dependent problems. Instead, we propose to use a piecewise parabolic approximation of the interface, resulting in a class of piecewise parabolic interface calculation (PPIC) methods. In particular, we introduce the parabo…
Provably convergent Newton-Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics
