Optimal second order diagonally implicit SSP Runge--Kutta methods

TL;DR

This paper introduces a new analytical approach to identify the unique optimal second order diagonally implicit SSP Runge--Kutta methods, confirming the optimality of the iterated implicit midpoint rule within this class.

Contribution

It provides the first general proof of the optimality of the iterated implicit midpoint rule for second order diagonally implicit SSP Runge--Kutta methods.

Findings

Proves the uniqueness of the optimal methods in the class.

Confirms the optimality of the iterated implicit midpoint rule.

Introduces a new analytical framework for SSP methods.

Abstract

Optimal Strong Stability Preserving (SSP) Runge--Kutta methods has been widely investegated in the last decade and many open conjectures have been formulated. The iterated implicit midpoint rule has been observed numerically optimal in large classes of second order methods, and was proven to be optimal for some small cases, but no general proof was known so far to show its optimality. In this paper we show a new approach to analytically investigate this problem and determine the unique optimal methods in the class of second order diagonally implicit Runge--Kutta methods.