Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Total-variation-diminishing time discretizations
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Numerical Analysis
Representations of Runge-Kutta Methods and Strong Stability Preserving Methods
SIAM Journal on Numerical Analysis
Optimal Strong-Stability-Preserving Time-Stepping Schemes with Fast Downwind Spatial Discretizations
Journal of Scientific Computing
Stepsize Conditions for General Monotonicity in Numerical Initial Value Problems
SIAM Journal on Numerical Analysis
A Numerical Study of Diagonally Split Runge---Kutta Methods for PDEs with Discontinuities
Journal of Scientific Computing
Strong stability of singly-diagonally-implicit Runge--Kutta methods
Applied Numerical Mathematics
Optimal implicit strong stability preserving Runge--Kutta methods
Applied Numerical Mathematics
High Order Strong Stability Preserving Time Discretizations
Journal of Scientific Computing
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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 semidiscretizations. 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.