Convergence analysis of multi-level spectral deferred corrections

TL;DR

This paper provides a theoretical convergence proof for multi-level spectral deferred corrections (MLSDC), a hierarchical extension of SDC methods for solving ODEs, including convergence rates and limitations compared to SDC.

Contribution

It offers the first rigorous convergence analysis of MLSDC, establishing conditions and rates, and compares its efficiency to standard SDC methods.

Findings

MLSDC converges under certain conditions.

Convergence rate depends on time-step size.

MLSDC has limitations compared to SDC in some scenarios.

Abstract

The spectral deferred correction (SDC) method is class of iterative solvers for ordinary differential equations (ODEs). It can be interpreted as a preconditioned Picard iteration for the collocation problem. The convergence of this method is well-known, for suitable problems it gains one order per iteration up to the order of the quadrature method of the collocation problem provided. This appealing feature enables an easy creation of flexible, high-order accurate methods for ODEs. A variation of SDC are multi-level spectral deferred corrections (MLSDC). Here, iterations are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. While there are several numerical examples which show its capabilities and efficiency, a theoretical convergence proof is still missing. This paper addresses this issue. A proof of the convergence of MLSDC, including the determination of the convergence rate in the time-step size, will be given and the results of the theoretical analysis will be numerically demonstrated. It turns out that there are restrictions for the advantages of this method over SDC regarding the convergence rate.