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
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods
SIAM Journal on Numerical Analysis
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
A Numerical Study of Diagonally Split Runge---Kutta Methods for PDEs with Discontinuities
Journal of Scientific Computing
Highly Efficient Strong Stability-Preserving Runge-Kutta Methods with Low-Storage Implementations
SIAM Journal on Scientific Computing
Strong stability of singly-diagonally-implicit Runge--Kutta methods
Applied Numerical Mathematics
High Order Strong Stability Preserving Time Discretizations
Journal of Scientific Computing
Characterizing Strong Stability Preserving Additive Runge-Kutta Methods
Journal of Scientific Computing
Strong-stability-preserving 3-stage Hermite-Birkhoff time-discretization methods
Applied Numerical Mathematics
SIAM Journal on Scientific Computing
Parallel High-Order Integrators
SIAM Journal on Scientific Computing
On the Linear Stability of the Fifth-Order WENO Discretization
Journal of 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
Strong Stability for Runge---Kutta Schemes on a Class of Nonlinear Problems
Journal of Scientific Computing
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Strong stability preserving (SSP) time discretizations were developed for use with spatial discretizations of partial differential equations that are strongly stable under forward Euler time integration. SSP methods preserve convex boundedness and contractivity properties satisfied by forward Euler, under a modified timestep restriction. We turn to implicit Runge-Kutta methods to alleviate this timestep restriction, and present implicit SSP Runge-Kutta methods which are optimal in the sense that they preserve convex boundedness properties under the largest timestep possible among all methods with a given number of stages and order of accuracy. We consider methods up to order six (the maximal order of SSP Runge-Kutta methods) and up to eleven stages. The numerically optimal methods found are all diagonally implicit, leading us to conjecture that optimal implicit SSP Runge-Kutta methods are diagonally implicit. These methods allow a larger SSP timestep, compared to explicit methods of the same order and number of stages. Numerical tests verify the order and the SSP property of the methods.