Efficient high-order two-derivative DIRK methods with optimized phase errors

TL;DR

This paper develops new high-order two-derivative diagonally implicit Runge--Kutta schemes with optimized phase errors, demonstrating improved accuracy and efficiency for solving ODEs and PDEs.

Contribution

It introduces new families of 2- and 3-stage high-order TDDIRK schemes with optimized phase errors and provides convergence and stability analysis.

Findings

New 2-stage fourth-order TDDIRK schemes outperform existing methods.

New 2-stage and 3-stage fifth-order TDDIRK schemes show improved accuracy.

Numerical experiments confirm enhanced efficiency of the proposed schemes.

Abstract

This work constructs and analyzes new efficient high-order two-derivative diagonally implicit Runge--Kutta (TDDIRK) schemes with optimized phase errors. Specifically, we present a convergence result for TDDIRK methods and investigate their optimized phase errors and linear stability analysis. Based on these, we derive new families of 2-stage fourth-order, 2-stage fifth-order, and 3-stage fifth-order TDDIRK schemes. Finally, we provide numerical experiments at both the ODE and PDE levels to demonstrate the accuracy and efficiency of these new schemes compared to known DIRK schemes in the literature.