题目：Monotonicity and discrete maximum principle in high order accurate schemes for diffusion operators

报告人: Xiangxiong Zhang

Purdue University 

时间： 2019年6月28日  3:30-4:30

地点： 五教 5207

摘要： 

In many applications modelling diffusion, it is desired for numerical schemes to have discrete maximum principle and bound-preserving (or positivity preserving) properties. Monotonicity of numerical schemes is a convenient tool to ensure these properties. For instance, it is well know that second order centered difference and piecewise linear finite element method on triangular meshes for the Laplacian operator has a monotone stiffness matrix, i.e., the inverse of the stiffness matrix has non-negative entries because the stiffness matrix is an M-matrix. Most high order accurate schemes simply do not satisfy the discrete maximum principle. In this talk, I will first review a few known high order schemes satisfying monotonicity for the Laplacian in the literature then present a new result: the finite difference implementation of continuous finite element method with tensor product of quadratic polynomial basis is monotone thus satisfies the discrete maximum principle for the variable coefficient Poisson equation. Such a scheme can be proven to be fourth order accurate. This is the first time that a high order accurate scheme that is proven to satisfy the discrete maximum principle for a variable coefficient diffusion operator. Applications including compressible Navier-Stokes equations will also be discussed.