Optimal second order diagonally implicit SSP Runge--Kutta methods
Provides a theoretical proof for optimal numerical methods in time integration, benefiting computational scientists working on hyperbolic PDEs.
The paper proves the optimality of the iterated implicit midpoint rule among second-order diagonally implicit SSP Runge-Kutta methods, resolving a long-standing conjecture.
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.