$h$ and $hp$-adaptive Interpolation by Transformed Snapshots for Parametric and Stochastic Hyperbolic PDEs
For computational scientists solving parametric hyperbolic PDEs, this work provides adaptive methods that overcome limitations of existing snapshot-based approaches for problems with parameter-dependent shock locations.
The paper addresses the challenge of approximating solutions to parametric/stochastic hyperbolic PDEs with shock singularities. It introduces h- and hp-adaptive methods based on transformed snapshots, including a tensorized adaptation approach to handle local changes in jump set structure, achieving improved approximation efficiency.
The numerical approximation of solutions of parametric or stochastic hyperbolic PDEs is still a serious challenge. Because of shock singularities, most methods from the elliptic and parabolic regime, such as reduced basis methods, POD or polynomial chaos expansions, show a poor performance. Recently, Welper [Interpolation of functions with parameter dependent jumps by transformed snapshots. SIAM Journal on Scientific Computing, 39(4):A1225-A1250, 2017] introduced a new approximation method, based on the alignment of the jump sets of the snapshots. If the structure of the jump sets changes with parameter, this assumption is too restrictive. However, these changes are typically local in parameter space, so that in this paper, we explore $h$ and $hp$-adaptive methods to resolve them. Since local refinements do not scale to high dimensions, we introduce an alternative "tensorized" adaption method.