A Generalization of Prefactored Compact Schemes for Advection Equations

TL;DR

This paper introduces a generalized prefactorization method for compact schemes in advection equations that reduces computational load while maintaining high-order accuracy, applicable in predictor-corrector frameworks.

Contribution

It proposes a novel prefactorization technique for compact schemes that simplifies computations without sacrificing accuracy, applicable up to sixteenth order.

Findings

Preserves order of accuracy in linear and nonlinear advection tests.

Reduces computational complexity by avoiding matrix inversion.

Demonstrates increased efficiency in numerical simulations.

Abstract

A generalized prefactorization of compact schemes aimed at reducing the stencil and improving the computational efficiency is proposed here in the framework of transport equations. By the prefactorization introduced here, the computational load associated with inverting multi-diagonal matrices is avoided, while the order of accuracy is preserved. The prefactorization can be applied to any centered compact difference scheme with arbitrary order of accuracy (results for compact schemes of up to sixteenth order of accuracy are included in the study). One notable restriction is that the proposed schemes can be applied in a predictor-corrector type marching scheme framework. Two test cases, associated with linear and nonlinear advection equations, respectively, are included to show the preservation of the order of accuracy and the increase of the computational efficiency of the prefactored compact schemes.