Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes

TL;DR

This paper explores the integration of relaxation deferred correction methods with residual distribution schemes, establishing their connection to Runge-Kutta methods and developing entropy conservative/dissipative schemes for hyperbolic problems.

Contribution

It demonstrates the link between DeC schemes and RK methods, and introduces entropy conservative/dissipative RDeC methods within the residual distribution framework.

Findings

DeC schemes are equivalent to convex combinations of explicit Euler steps.

RDeC methods can be constructed to be entropy conservative or dissipative.

The approach improves high-order explicit schemes for hyperbolic problems.

Abstract

In [1] is proposed a simplified DeC method, that, when combined with the residual distribution (RD) framework, allows to construct a high order, explicit FE scheme with continuous approximation avoiding the inversion of the mass matrix for hyperbolic problems. In this paper, we close some open gaps in the context of deferred correction (DeC) and their application within the RD framework. First, we demonstrate the connection between the DeC schemes and the RK methods. With this knowledge, DeC can be rewritten as a convex combination of explicit Euler steps, showing the connection to the strong stability preserving (SSP) framework. Then, we can apply the relaxation approach introduced in [2] and construct entropy conservative/dissipative DeC (RDeC) methods, using the entropy correction function proposed in [3]. [1] R. Abgrall. High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. Journal of Scientific Computing, 73(2):461--494, Dec 2017. [2] D. Ketcheson. Relaxation Runge--Kutta methods: Conservation and stability for inner-product norms. SIAM Journal on Numerical Analysis, 57(6):2850--2870, 2019. [3] R. Abgrall. A general framework to construct schemes satisfying additional conservation relations. application to entropy conservative and entropy dissipative schemes. Journal of Computational Physics, 372:640--666, 2018.