Related papers: A convergent Lagrangian discretization for a nonli…
In this work we present a mass conservative numerical scheme for two-phase flow in porous media. The model for flow consists on two fully coupled, non-linear equations: a degenerate parabolic equation and an elliptic equation. The proposed…
We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation…
The Landau-Lifshitz-Gilbert (LLG) equation, regarded as a gradient flow with manifold constraint, is the fundamental model describing magnetization dynamics in ferromagnetic materials. It is well known that the normalized tangent plane…
A nonlinear Schr\"odinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schr\"odinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show…
A new Lagrangian particle method for solving Euler equations for compressible inviscid fluid or gas flows is proposed. Similar to smoothed particle hydrodynamics (SPH), the method represents fluid cells with Lagrangian particles and is…
We present reliable a posteriori estimators for some fully discrete schemes applied to nonlinear systems of hyperbolic conservation laws in one space dimension with strictly convex entropy. The schemes are based on a method of lines…
We consider a fully discrete scheme for nonlinear stochastic partial differential equations with non-globally Lipschitz coefficients driven by multiplicative noise in a multi-dimensional setting. Our method uses a polynomial based spectral…
We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad…
When expressed in Lagrangian variables, the equations of motion for compressible (barotropic) fluids have the structure of a classical Hamiltonian system in which the potential energy is given by the internal energy of the fluid. The…
We present a numerical method for approximating the solutions of degenerate parabolic equations with a formal gradient flow structure. The numerical method we propose preserves at the discrete level the formal gradient flow structure,…
An implicit Euler finite-volume scheme for general cross-diffusion systems with volume-filling constraints is proposed and analyzed. The diffusion matrix may be nonsymmetric and not positive semidefinite, but the diffusion system is assumed…
Explicit, unconditionally stable, high-order schemes for the approximation of some first- andsecond-order linear, time-dependent partial differential equations (PDEs) are proposed.The schemes are based on a weak formulation of a…
Using uniform global Carleman estimates for discrete elliptic and semi-discrete hyperbolic equations, we study Lipschitz and logarithmic stability for the inverse problem of recovering a potential in a semi-discrete wave equation,…
We develop a fully discrete, semi-implicit mixed finite element method for approximating solutions to a class of fourth-order stochastic partial differential equations (SPDEs) with non-globally Lipschitz and non-monotone nonlinearities,…
We consider the study of a numerical scheme for an initial- and Dirichlet boundary- value problem for a nonlinear Schr\"odinger equation. We approximate the solution using a, local (non-uniform) two level scheme in time (see C. Besse [6]…
We develop a numerical strategy to solve multi-dimensional Poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts. The method is an extension of the multiresolution finite volume scheme used…
In this work, we propose a new semi-Lagrangian (SL) finite difference scheme for nonlinear advection-diffusion problems. To ensure conservation, which is fundamental for achieving physically consistent solutions, the governing equations are…
We present a novel framework based on semi-bounded spatial operators for analyzing and discretizing initial boundary value problems on moving and deforming domains. This development extends an existing framework for well-posed problems and…
This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for…
We study a general convergence theory for the numerical solutions of compressible viscous and electrically conducting fluids with a focus on numerical schemes that preserve the divergence free property of magnetic field exactly. Our…