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

TL;DR

This paper introduces an adjoint-guided adaptive mesh refinement method for linear hyperbolic PDEs, improving targeted refinement efficiency in large domains by focusing on waves influencing a specific area, implemented in Clawpack.

Contribution

It develops and demonstrates an adjoint-based refinement technique integrated into Clawpack, enhancing the precision of wave refinement for targeted regions in PDE simulations.

Findings

Adjoint method improves refinement accuracy for target areas.

Compared to existing AMR, the adjoint approach reduces unnecessary computations.

Implementation in Clawpack shows practical feasibility and benefits.

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 be used to track discontinuities in the solution by selectively refining around those areas. 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 how this method has been implemented in AMRClaw. Examples for linear acoustics equations are presented, and a computational performance analysis is conducted. The adjoint method is compared to AMR methods already available in the AMRClaw software, and the resulting advantages and disadvantages are discussed. The code for the examples presented is archived on Github.