Space-time adaptive methods for parabolic evolution equations

TL;DR

This paper introduces adaptive integral equation-based solvers for various parabolic and fluid dynamics equations in 2D, utilizing space-time refinement techniques to efficiently capture complex evolving features.

Contribution

It develops novel adaptive space-time methods with quadtree refinement for solving parabolic PDEs and fluid equations, enhancing efficiency and accuracy.

Findings

Methods effectively track complex solution features.

Numerical examples demonstrate robustness and efficiency.

Adaptive refinement improves computational performance.

Abstract

We present a family of integral equation-based solvers for the heat equation, reaction-diffusion systems, the unsteady Stokes equation and the incompressible Navier-Stokes equations in two space dimensions. Our emphasis is on the development of methods that can efficiently follow complex solution features in space-time by refinement and coarsening at each time step on an adaptive quadtree. For simplicity, we focus on problems posed in a square domain with periodic boundary conditions. The performance and robustness of the methods are illustrated with several numerical examples.