Very High-Order A-stable Stiffly Accurate Diagonally Implicit Runge-Kutta Methods with Error Estimators

TL;DR

This paper introduces new high-order A-stable diagonally implicit Runge-Kutta methods with embedded error estimators, improving accuracy and stability for stiff differential equations and DAEs, using advanced optimization techniques.

Contribution

It presents the highest order A-stable DIRK schemes up to order eight, including schemes with improved stability and accuracy features, developed through complex numerical optimization.

Findings

New schemes demonstrate high accuracy and stability on diverse problems.

Eighth-order schemes require advanced optimization for system solutions.

Schemes with stage order two improve accuracy for stiff problems.

Abstract

A numerical search approach is used to design high-order diagonally implicit Runge-Kutta (DIRK) schemes equipped with embedded error estimators, some of which have identical diagonal elements (SDIRK) and explicit first stage (ESDIRK). In each of these classes, we present new A-stable schemes of order six (the highest order of previously known A-stable DIRK-type schemes) up to order eight. For each order, we include one scheme that is only A-stable as well as schemes that are L-stable, stiffly accurate, and/or have stage order two. The latter types require more stages, but give better convergence rates for differential-algebraic equations (DAEs), and those which have stage order two give better accuracy for moderately stiff problems. The development of the eighth-order schemes requires, in addition to imposing A-stability, finding highly accurate numerical solutions for a system of 200 equations in over 100 variables, which is accomplished via a combination of global and local optimization strategies. The accuracy, stability, and adaptive stepsize control of the schemes are demonstrated on diverse problems.