Bound-preserving high order schemes

For the initial value problem of scalar conservation laws, a bound-preserving property is desired for numerical schemes in many applications. Traditional methods to enforce a discrete maximum principle by defining the extrema as those of grid point values in finite difference schemes or cell averages in finite volume schemes usually result in an accuracy degeneracy to second order around smooth extrema. On the other hand, successful and popular high order accurate schemes do not satisfy a strict bound-preserving property. We review two approaches for enforcing the bound-preserving property in high order schemes. The first one is a general framework to design a simple and efficient limiter for finite volume and discontinuous Galerkin schemes without destroying high order accuracy. The second one is a bound-preserving flux limiter, which can be used on high order finite difference, finite volume and discontinuous Galerkin schemes.