Bound-Preserving High Order Finite Volume Schemes for Conservation Laws and Convection-Diffusion Equations

Abstract

Finite volume schemes evolve cell averages based on high order reconstructions to solve hyperbolic conservation laws and convection-diffusion equations. The design of the reconstruction procedure is crucial for the stability of the finite volume schemes. Various reconstruction procedures, such as total variation diminishing (TVD), total variation bounded (TVB), essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstructions have been developed in the literature to obtain non-oscillatory and high order finite volume schemes. However, it is a challenge to design strictly bound-preserving finite volume schemes which are genuinely high order accurate, including at smooth extrema. These include maximum-principle-preserving schemes for scalar conservation laws and convection-diffusion equations, and positivity-preserving (for relevant physical quantities such as density, pressure or water height) for systems. In this presentation we survey strategies in the recent literature to design high order bound-preserving finite volume schemes, including a general framework in constructing high order bound-preserving finite volume schemes for scalar and systems of hyperbolic conservation laws through a simple scaling limiter and a convex combination argument based on first order bound-preserving building blocks, and a non-standard finite volume scheme which evolves the so-called “double cell averages” for solving convection-diffusion equations which can maintain the bound-preserving property and high order accuracy simultaneously.