Location
MC 6460
Candidate
Dongze Li | Applied Math, University of Waterloo
Title
High-order provably-stable scheme on space-time curvilinear coordinates with shock-tracking capability
Abstract
Robust, efficient, and stable numerical methods are essential to support the design of next-generation aircraft. High-order methods can result in efficient and accurate simulations and can provide high-fidelity solutions for aerodynamic problems. Despite these benefits, high-order methods lack robustness, for example, for flows with shock. There exist many approaches to mitigate the lack of robustness, however, they face drawbacks such as reduction to first order around shocks and increased computational cost.
Recently, Zahr and coauthors introduced a new approach, referred to as implicit shock tracking, that results in a mesh that has interfaces aligned with shocks, and therefore removes many of the difficulties faced by high-order methods for shock problems. Unfortunately, this approach does not guarantee stability. However, the summation-by-parts (SBP) framework provides a systematic approach for developing provably stable schemes. In particular, this framework has been used to develop high-order provably-stable schemes for the unsteady 3-D Euler and Navier-Stokes equations.
In this presentation, I will start by presenting SBP operators in computational coordinates and their properties, extending them to 2D computational domain, and then to arbitrarily physical space-time domains. Then the space-time discretization of the linear advection equation and Burger’s equation in curvilinear coordinates are presented and proven to be stable. Finally, some results are presented numerically verifying the accuracy and stability properties of the schemes.