Comparison of adaptive multiresolution and adaptive mesh refinement applied to simulations of the compressible Euler equations

TL;DR

This paper compares adaptive multiresolution and adaptive mesh refinement methods for solving the compressible Euler equations, analyzing their efficiency, accuracy, and memory usage across various test cases.

Contribution

It provides a detailed benchmark comparison of MR and AMR methods, highlighting their relative efficiencies and convergence behaviors in PDE simulations.

Findings

Both methods show similar CPU time compression with more refinement levels.

MR is more memory-efficient and slightly more accurate than AMR.

AMR has lower absolute overhead despite similar convergence trends.

Abstract

We present a detailed comparison between two adaptive numerical approaches to solve partial differential equations (PDEs), adaptive multiresolution (MR) and adaptive mesh refinement (AMR). Both discretizations are based on finite volumes in space with second order shock-capturing, and explicit time integration either with or without local time-stepping. The two methods are benchmarked for the compressible Euler equations in Cartesian geometry. As test cases a 2D Riemann problem, Lax-Liu 6, and a 3D ellipsoidally expanding shock wave have been chosen. We compare and assess their computational efficiency in terms of CPU time and memory requirements. We evaluate the accuracy by comparing the results of the adaptive computations with those obtained with the corresponding FV scheme using a regular fine mesh. We find that both approaches yield similar trends for CPU time compression for increasing number of refinement levels. MR exhibits more efficient memory compression than AMR and shows slightly enhanced convergence; however, a larger absolute overhead is measured for the tested codes.