A new efficient explicit Deferred Correction framework: analysis and applications to hyperbolic PDEs and adaptivity

TL;DR

This paper introduces an efficient explicit Deferred Correction framework that reduces computational costs and enhances applicability to hyperbolic PDEs and adaptive methods, with stability analysis and broad benchmarking.

Contribution

It proposes a novel interpolation-based modification to explicit DeC methods, improving efficiency and enabling advanced applications in PDEs and adaptivity.

Findings

Reduced computational cost through interpolation processes

Stable in certain cases despite efficiency improvements

Effective in ODE and PDE benchmark tests

Abstract

The Deferred Correction (DeC) is an iterative procedure, characterized by increasing accuracy at each iteration, which can be used to design numerical methods for systems of ODEs. The main advantage of such framework is the automatic way of getting arbitrarily high order methods, which can be put in Runge--Kutta (RK) form. The drawback is the larger computational cost with respect to the most used RK methods. To reduce such cost, in an explicit setting, we propose an efficient modification: we introduce interpolation processes between the DeC iterations, decreasing the computational cost associated to the low order ones. We provide the Butcher tableaux of the new modified methods and we study their stability, showing that in some cases the computational advantage does not affect the stability. The flexibility of the novel modification allows nontrivial applications to PDEs and construction of adaptive methods. The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.