A multiresolution adaptive wavelet method for nonlinear partial differential equations

TL;DR

This paper introduces a multiresolution wavelet algorithm for efficiently solving nonlinear PDEs with adaptive grid refinement, data compression, and explicit error control, suitable for complex multi-scale problems.

Contribution

It develops a novel multiresolution wavelet method that adaptively discretizes PDEs with guaranteed accuracy and dynamic grid adaptation, improving computational efficiency for multi-scale problems.

Findings

Demonstrates high accuracy in solving PDEs with wavelet-based discretization.

Shows effective adaptive grid refinement maintaining solution accuracy.

Achieves significant data compression while controlling errors.

Abstract

The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to solve partial differential equations (PDEs) with features evolving on a wide range of spatial and temporal scales. To meet these challenges, we present a multiresolution wavelet algorithm to solve PDEs with significant data compression and explicit error control. We discretize in space by projecting fields and spatial derivative operators onto wavelet basis functions. We provide error estimates for the wavelet representation of fields and their derivatives. Then, our estimates are used to construct a sparse multiresolution discretization which guarantees the prescribed accuracy. Additionally, we embed a predictor-corrector procedure within the temporal integration to dynamically adapt the computational grid and maintain the accuracy of the solution of the PDE as it evolves. We present examples to highlight the accuracy and adaptivity of our approach.