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
Contractivity of Runge-Kutta methods with respect to forcing terms
Applied Numerical Mathematics
SIAM Journal on Numerical Analysis
A note on unconditional maximum norm contractivity of diagonally split Runge-Kutta methods
SIAM Journal on Numerical Analysis
Unconditional Contractivity in the Maximum Norm of Diagonally Split Runge--Kutta Methods
SIAM Journal on Numerical Analysis
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
Positivity of Runge-Kutta and diagonally split Runge-Kutta methods
Applied Numerical Mathematics - Selected papers on eighth conference on the numerical treatment of differential equations 1-5 September 1997, Alexisbad, Germany
A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods
SIAM Journal on Numerical Analysis
Two Barriers on Strong-Stability-Preserving Time Discretization Methods
Journal of Scientific Computing
Non-linear evolution using optimal fourth-order strong-stability-preserving Runge-Kutta methods
Mathematics and Computers in Simulation - Nonlinear waves: computation and theory II
On Strong Stability Preserving Time Discretization Methods
Journal of Scientific Computing
Representations of Runge-Kutta Methods and Strong Stability Preserving Methods
SIAM Journal on Numerical Analysis
On High Order Strong Stability Preserving Runge---Kutta and Multi Step Time Discretizations
Journal of Scientific Computing
Stepsize Conditions for General Monotonicity in Numerical Initial Value Problems
SIAM Journal on Numerical Analysis
Strong stability of singly-diagonally-implicit Runge--Kutta methods
Applied Numerical Mathematics
Optimal implicit strong stability preserving Runge--Kutta methods
Applied Numerical Mathematics
Optimal Explicit Strong-Stability-Preserving General Linear Methods
SIAM Journal on Scientific Computing
Step Sizes for Strong Stability Preservation with Downwind-Biased Operators
SIAM Journal on Numerical Analysis
Strong Stability Preserving Two-step Runge-Kutta Methods
SIAM Journal on Numerical Analysis
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Diagonally split Runge---Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge---Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) time-stepping community which is generally concerned with computing oscillation-free numerical solutions of PDEs. Using a variety of numerical test problems, we show that although second- and third-order unconditionally contractive DSRK methods do preserve the strong stability property for all time step-sizes, they suffer from order reduction at large step-sizes. Indeed, for time-steps larger than those typically chosen for explicit methods, these DSRK methods behave like first-order implicit methods. This is unfortunate, because it is precisely to allow a large time-step that we choose to use implicit methods. These results suggest that unconditionally contractive DSRK methods are limited in usefulness as they are unable to compete with either the first-order backward Euler method for large step-sizes or with Crank---Nicolson or high-order explicit SSP Runge---Kutta methods for smaller step-sizes.We also present stage order conditions for DSRK methods and show that the observed order reduction is associated with the necessarily low stage order of the unconditionally contractive DSRK methods.