Investigation of Stabilization Methods for Multi-Dimensional Summation-by-parts Discretizations of the Euler Equations

摘要

We present an extensible Julia-based solver for the Euler equations that uses a summation-by-parts (SBP) discretization on unstructured triangular grids. While SBP operators have been used for tensor-product discretizations for some time, they have only recently been extended to simplices. Here we investigate the accuracy and stability properties of simplex-based SBP discretizations of the Euler equations. Non-linear stabilization is a particular concern in this context, because SBP operators are nearly skew-symmetric. We consider an edge-based stabilization method, which has previously been used for advection-diffusion-reaction problems and the Oseen equations, and apply it to the Euler equations. Additionally, we discuss how the development of our software has been facilitated by the use of Julia, a new, fast, dynamic programming language designed for technical computing. By taking advantage of Julia's unique capabilities, code that is both efficient and generic can be written, enhancing the extensibility of the solver.