Phase error analysis of implicit Runge-Kutta methods: Introducing new classes of minimal dissipation low dispersion high order schemes

TL;DR

This paper analyzes the phase error of Gauss-Legendre implicit Runge-Kutta methods, introduces new classes of schemes with minimal dissipation and low dispersion, and provides an algorithm for designing more accurate stable methods for larger time steps.

Contribution

It introduces a novel algorithm to design implicit Runge-Kutta schemes with optimized dissipation and dispersion properties for better accuracy at larger time steps.

Findings

Gauss-Legendre methods have minimal dissipation and high dispersive order.

New schemes reduce phase error at larger CFL numbers.

Numerical tests confirm improved accuracy of the proposed schemes.

Abstract

In current research, we analyse dissipation and dispersion characteristics of most accurate two and three stage Gauss-Legendre implicit Runge-Kutta (R-K) methods. These methods, known for their $A$-stability and immense accuracy, are observed to carry minimum dissipation error along with highest possible dispersive order in their respective classes. We investigate to reveal that these schemes are inherently optimized to carry low phase error only at small wavenumber. As larger temporal step size is imperative in conjunction with implicit R-K methods for physical problems, we interpret to derive a class of minimum dissipation and optimally low dispersion implicit R-K schemes. Schemes thus obtained by cutting down amplification error and maximum reduction of weighted phase error, suggest better accuracy for relatively bigger CFL number. Significantly, we are able to outline an algorithm that can be used to design stable implicit R-K methods for suitable time step with better accuracy virtues. The algorithm is potentially generalizable for implicit R-K class of methods. As we focus on two and three stage schemes a comprehensive comparison is carried out using numerical test cases.