Step Sizes for Strong Stability Preservation with Downwind-biased Operators

TL;DR

This paper analyzes the limits of step sizes in strong stability preserving methods using downwind-biased operators, showing bounds for multistep and Runge--Kutta methods and constructing high-coefficient second order methods.

Contribution

It establishes theoretical bounds on the SSP coefficients for downwind-biased methods and introduces new second order Runge--Kutta methods with arbitrarily large SSP coefficients.

Findings

Downwind SSP coefficient for linear multistep methods of order > 1 is at most 2.

Explicit Runge--Kutta methods have SSP coefficient at most equal to the number of stages.

Second order Runge--Kutta methods can have unbounded SSP coefficients.

Abstract

Strong stability preserving (SSP) integrators for initial value ODEs preserve temporal monotonicity solution properties in arbitrary norms. All existing SSP methods, including implicit methods, either require small step sizes or achieve only first order accuracy. It is possible to achieve more relaxed step size restrictions in the discretization of hyperbolic PDEs through the use of both upwind- and downwind-biased semi-discretizations. We investigate bounds on the maximum SSP step size for methods that include negative coefficients and downwind-biased semi-discretizations. We prove that the downwind SSP coefficient for linear multistep methods of order greater than one is at most equal to two, while the downwind SSP coefficient for explicit Runge--Kutta methods is at most equal to the number of stages of the method. In contrast, the maximal downwind SSP coefficient for second order Runge--Kutta methods is shown to be unbounded. We present a class of such methods with arbitrarily large SSP coefficient and demonstrate that they achieve second order accuracy for large CFL number.