Location
MC 5501
Candidate
Dongze Li | Applied Mathematics, University of Waterloo
Title
Entropy-stable summation-by-parts shock tracking method for compressible the Euler equations
Abstract
Robust, efficient, and stable numerical methods are essential for designing next-generation aircraft. High-order methods provide efficient and accurate simulations with high-fidelity solutions for aerodynamic problems. However, these methods lack robustness, especially in the presence of shocks, leading to spurious oscillations and instabilities that can cause numerical solvers to crash. To overcome these challenges, several stabilization techniques have been developed. Limiting strategies and ENO/WENO schemes reduce the high-order representation within elements. Alternatives like adding artificial viscosity and employing local h/p mesh refinement are also used. Despite these efforts, many of these methods degrade to first-order accuracy near shocks and increase computational costs.
Developing high-order, provably stable schemes for nonlinear conservation laws is challenging. The summation-by-parts (SBP) framework provides a discrete analog of integration-by-parts, ensuring stability by mimicking continuous properties. When combined with simultaneous approximation terms (SATs), the SBP framework becomes even more powerful by weakly enforcing boundary conditions and inter-element coupling, leading to provably stable schemes. The SBP-SAT framework is versatile and can solve partial differential equations (PDEs) on complex domains with general element types. However, it is insufficient for general nonlinear conservation laws. Recent advancements have combined SBP-SAT with Tadmor’s low-order entropy stable schemes, resulting in entropy-stable schemes that have been successfully applied to the compressible Euler and Navier-Stokes equations.
Recently, Persson and Zahr introduced a method to align mesh interfaces with shock surfaces. In this approach, the conservation law is discretized without knowledge of the shock's location. Then, an optimization problem adjusts the mesh to align with the shock, while finding the corresponding solution. This approach is advantageous as it can be applied to any discretization allowing inter-element discontinuities, ensuring that high-order accuracy is retained even for shocked problems.
In this project, we aim to combine implicit shock tracking with the SBP framework to develop high-order, provably-stable schemes for feature tracking in complex geometries, focusing on space-time curvilinear SBP operators for the compressible Euler equations. The goal is to accurately resolve not only shocks but more general discontinuous features like the shadow zone interface in ice-accretion problems.