报告题目：Monotonicity and discrete maximum principle in high order accurateschemes for diffusion operators
报告摘要：In many applications modelling diffusion, it is desired for numericalschemes to have discrete maximum principle and bound-preserving (or positivitypreserving) properties. Monotonicity of numerical schemes is a convenient toolto ensure these properties. For instance, it is well know that second ordercentered difference and piecewise linear finite element method on triangularmeshes for the Laplacian operator has a monotone stiffness matrix, i.e., theinverse of the stiffness matrix has non-negative entries because the stiffnessmatrix is an M-matrix. Most high order accurate schemes simply do not satisfythe discrete maximum principle. In this talk, I will first review a few knownhigh order schemes satisfying monotonicity for the Laplacian in the literaturethen present a new result: the finite difference implementation of continuousfinite element method with tensor product of quadratic polynomial basis ismonotone thus satisfies the discrete maximum principle for the variablecoefficient Poisson equation. Such a scheme can be proven to be fourth orderaccurate. This is the first time that a high order accurate scheme that isproven to satisfy the discrete maximum principle for a variable coefficientdiffusion operator. Applications including compressible Navier-Stokes equationswill also be discussed.
报告人简介：Xiangxiong has been an assistant professor of mathematics at PurdueUniversity since 2014. Before that, he was a postdoc at math department at MITfrom 2011 to 2014. He got his Ph.D. in mathematics at Brown University in 2011.His research interests are numerical analysis and scientific computingincluding high order accurate numerical methods for PDEs and optimizationalgorithms.