Fully adaptive multiresolution finite volume schemes for conservation laws

TLDR

Multiresolution decompositions were introduced by Harten to accelerate flux evaluation via adaptive computation, but the approach still represents the solution on the finest grid each step, limiting memory and time savings. The paper develops and analyzes fully adaptive multiresolution schemes that represent and compute the solution on a dynamically evolved adaptive grid, addressing the challenge of accurately computing fluxes without full fine‑grid cell averages. The authors propose, analyze, and compare several solutions for accurate flux computation on adaptive grids, evaluating them by accuracy and computational complexity. The comparative analysis demonstrates varying trade‑offs between accuracy and computational complexity among the proposed flux‑computation methods.

Abstract

The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity.