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We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics…
Stochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton-Jacobi-Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the…
A classical model for water-gas flows in porous media is considered. The degenerate coupled system of equations obtained by mass conservation is usually approximated by finite volume schemes in the oil reservoir simulations. The convergence…
A space-time adaptive method is presented for the reactive Euler equations describing chemically reacting gas flow where a two species model is used for the chemistry. The governing equations are discretized with a finite volume method and…
A cell-centered implicit-explicit updated Lagrangian finite volume scheme on unstructured grids is proposed for a unified first order hyperbolic formulation of continuum fluid and solid mechanics. The scheme provably respects the stiff…
We present a class of high order finite volume schemes for the solution of non-conservative hyperbolic systems that combines the one-step ADER-WENO finite volume approach with space-time adaptive mesh refinement (AMR). The resulting…
We introduce an extended discontinuous Galerkin discretization of hyperbolic-parabolic problems on multidimensional semi-infinite domains. Building on previous work on the one-dimensional case, we split the strip-shaped computational domain…
In this article, we have developed a higher order compact numerical method for variable coefficient parabolic problems with mixed derivatives. The finite difference scheme, presented here for two-dimensional domains, is based on fourth…
The fluid flow transport and hydrodynamic problems often take the form of hyperbolic systems of conservation laws. In this work we will present a new scheme of finite volume methods for solving these evolution equations. It is a family of…
In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the…
We propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. It is based on the original exploitation of a result related to…
This paper solves the discretised multiphase flow equations using tools and methods from machine-learning libraries. The idea comes from the observation that convolutional layers can be used to express a discretisation as a neural network…
An implicit Euler finite-volume scheme for a degenerate cross-diffusion system describing the ion transport through biological membranes is analyzed. The strongly coupled equations for the ion concentrations include drift terms involving…
We introduce a family of hybrid discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous…
This article presents a high order conservative flux optimization (CFO) finite element method for the elliptic diffusion equations. The numerical scheme is based on the classical Galerkin finite element method enhanced by a flux…
In this paper, we study the convergence analysis for a robust stochastic structure-preserving Lagrangian numerical scheme in computing effective diffusivity of time-dependent chaotic flows, which are modeled by stochastic differential…
We prove optimal convergence rates for the discretization of a general second-order linear elliptic PDE with an adaptive vertex-centered finite volume scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54 (2016), pp.…
In this study, we propose a new scheme named as complete flux scheme (CFS) based on the finite volume method for solving singularly perturbed differential-difference equations (SPDDEs) of elliptic type. An alternate integral representation…
This article presents a new high-order accurate algorithm for finding a particular solution to a linear, constant-coefficient partial differential equation (PDE) by means of a convolution of the volumetric source function with the Green's…
The Osher-Chakrabarthy family of linear flux-modification schemes is considered. Improved lower bounds on the compression factors are provided, which suggest the viability of using the unlimited version. The LLF flux formula is combined…