Modern Perspectives in Applied Mathematics: Theory and Numerics of PDEs

Positivity-preserving high order schemes for convection dominated equations

Chi-Wang Shu

Brown University

We give a survey of our recent work with collaborators on the construction of uniformly high order accurate discontinuous Galerkin (DG) and weighted essentially non-oscillatory (WENO) finite volume (FV) and finite difference (FD) schemes which satisfy strict maximum principle for nonlinear scalar conservation laws, passive convection in incompressible flows, and nonlinear scalar convection-diffusion equations, and preserve positivity for density and pressure for compressible Euler systems. A general framework (for arbitrary order of accuracy) is established to construct a simple scaling limiter for the DG or FV method involving only evaluations of the polynomial solution at certain quadrature points. The bound preserving property is guaranteed for the first order Euler forward time discretization or strong stability preserving (SSP) high order time discretizations under suitable CFL condition. One remarkable property of this approach is that it is straightforward to extend the method to two and higher dimensions on arbitrary triangulations. We will emphasize recent developments including arbitrary equations of state, source terms, integral terms, shallow water equations, high order accurate finite difference positivity preserving schemes for Euler equations, and positivity-preserving high order finite volume scheme and piecewise linear DG scheme for convection-diffusion equations. Numerical tests demonstrating the good performance of the scheme will be reported.