Limiting and shock detection for discontinuous Galerkin solutions using multiwavelets

Abstract

Many areas such as climate modeling, shallow water equations, and computational fluid dynamics use nonlinear hyperbolic partial differential equations (PDE’s) to describe the behaviour of some unknown quantity. In general, the solutions of these equations contain shocks, or develop discontinuities. To solve these equations, various types of numerical methods can be used, such as finite difference, finite volume and finite element methods. In this master thesis, the discontinuous Galerkin method (DG) is used. To efficiently apply DG in case of discontinuous solutions, limiting techniques are used to reduce the spurious oscillations, that are developed in the discontinuous regions. Unfortunately, most of the limiters do not work well for higher order approximations, or multidimensional cases. Originally, the project focused on limiting DG solutions using multiwavelets. However, it is very hard to limit the solution using the multiwavelet decomposition. We did discover that the multiwavelet expansion is quite practical for shock detection. This report describes the pros and cons of multiwavelets as a limiter and shock detector for limiting DG.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc