High order implicit time integration schemes on multiresolution adaptive grids for stiff PDEs

TL;DR

This paper introduces high-order implicit Runge-Kutta schemes combined with multiresolution adaptive grids to efficiently solve stiff PDEs with localized fronts, significantly reducing computational costs.

Contribution

It develops a novel numerical strategy integrating multiresolution analysis with high-order implicit schemes for efficient, accurate simulation of unsteady, stiff PDEs on adaptive grids.

Findings

Demonstrates computational efficiency in complex physical scenarios

Achieves high accuracy with reduced computational resources

Handles highly unsteady problems with localized features

Abstract

We consider high order, implicit Runge-Kutta schemes to solve time-dependent stiff PDEs on dynamically adapted grids generated by multiresolution analysis for unsteady problems disclosing localized fronts. The multiresolution finite volume scheme yields highly compressed representations within a user-defined accuracy tolerance, hence strong reductions of computational requirements to solve large, coupled nonlinear systems of equations. SDIRK and RadauIIA Runge-Kutta schemes are implemented with particular interest in those with L-stability properties and accuracy-based time-stepping capabilities. Numerical evidence is provided of the computational efficiency of the numerical strategy to cope with highly unsteady problems modeling various physical scenarios with a broad spectrum of time and space scales.