Analysis and Performance Evaluation of Adjoint-guided Adaptive Mesh Refinement for Linear Hyperbolic PDEs Using Clawpack

Abstract

Adaptive mesh refinement (AMR) is often used when solving time-dependent partial differential equations using numerical methods. It enables time-varying regions of much higher resolution, which can selectively refine areas to track discontinuities in the solution. The open source Clawpack software implements block-structured AMR to refine around propagating waves in the AMRClaw package. For problems where the solution must be computed over a large domain but is only of interest in a small area, this approach often refines waves that will not impact the target area. We seek a method that enables the identification and refinement of only the waves that will influence the target area. Here we show that solving the time-dependent adjoint equation and using a suitable inner product allows for a more precise refinement of the relevant waves. We present the adjoint methodology in general and give details on the implementation of this method in AMRClaw. Examples and a computational performance analysis for linear acoustics equations are presented. The adjoint method is compared to AMR methods already available in AMRClaw, and the advantages and disadvantages are discussed. The approach presented here is implemented in Clawpack, in Version 5.6.1, and code for all examples presented is archived on Github.