Abstract

Solutions to many hyperbolic equations have convex invariant
regions, for example solutions to scalar conservation laws satisfy
maximum principle, solutions to compressible Euler equations
satisfy positivity-preserving property for density and internal
energy, etc. It is however a challenge to design schemes whose
solutions also honor such invariant regions. This is especially
the case for high order accurate schemes. In this talk we will
first survey strategies in the literature to design high order
bound-preserving schemes, including the general framework in
constructing high order bound-preserving finite volume and
discontinuous Galerkin schemes for scalar and systems of
hyperbolic equations through a simple scaling limiter and a
convex combination argument based on first order bound-preserving
building blocks, and various flux limiters to design high order
bound-preserving finite difference schemes. We will then discuss
a few recent developments, including high order bound-preserving
schemes for relativistic hydrodynamics, high order discontinuous
Galerkin Lagrangian schemes, and high order discontinuous Galerkin
methods for radiative transfer equations. Numerical tests
demonstrating the good performance of these schemes will be reported.