A recursive system-free single-step temporal discretization method for finite difference methods

TL;DR

This paper introduces a recursive, system-agnostic method for efficiently computing high-order temporal discretizations in finite difference methods, simplifying the calculation of complex tensor contractions in PDE solutions.

Contribution

It proposes a novel recursive Jacobian operator approach that streamlines high-order temporal discretization without system-specific tensor calculations.

Findings

Efficient computation of tensor contractions for high-order schemes.

System-agnostic approach applicable to various PDE systems.

Enhanced simplicity and efficiency in single-step temporal discretizations.

Abstract

Single-stage or single-step high-order temporal discretizations of partial differential equations (PDEs) have shown great promise in delivering high-order accuracy in time with efficient use of computational resources. There has been much success in developing such methods for finite volume method (FVM) discretizations of PDEs. The Picard Integral formulation (PIF) has recently made such single-stage temporal methods accessible for finite difference method (FDM) discretizations. PIF methods rely on the so-called Lax-Wendroff procedures to tightly couple spatial and temporal derivatives through the governing PDE system to construct high-order Taylor series expansions in time. Going to higher than third order in time requires the calculation of Jacobian-like derivative tensor-vector contractions of an increasingly larger degree, greatly adding to the complexity of such schemes. To that end, we present in this paper a method for calculating these tensor contractions through a recursive application of a discrete Jacobian operator that readily and efficiently computes the needed contractions entirely agnostic of the system of partial differential equations (PDEs) being solved.