A multi-level spectral deferred correction method

TL;DR

This paper introduces a multi-level spectral deferred correction (MLSDC) method for PDEs that reduces computational cost by employing hierarchical levels and various coarsening strategies, demonstrating significant efficiency gains in 3D problems.

Contribution

The paper develops and analyzes a multi-level SDC approach with novel coarsening strategies, improving efficiency over traditional SDC for PDE integration.

Findings

MLSDC achieves faster convergence than SDC in numerical tests.

Coarsening strategies significantly reduce computational cost.

MLSDC provides substantial savings in 3D PDE simulations.

Abstract

The spectral deferred correction (SDC) method is an iterative scheme for computing a higher-order collocation solution to an ODE by performing a series of correction sweeps using a low-order timestepping method. This paper examines a variation of SDC for the temporal integration of PDEs called multi-level spectral deferred corrections (MLSDC), where sweeps are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. Three different strategies to reduce the computational cost of correction sweeps on the coarser levels are examined: reducing the degrees of freedom, reducing the order of the spatial discretization, and reducing the accuracy when solving linear systems arising in implicit temporal integration. Several numerical examples demonstrate the effect of multi-level coarsening on the convergence and cost of SDC integration. In particular, MLSDC can provide significant savings in compute time compared to SDC for a three-dimensional problem.