ImplicitLNLMethods
Implicit and implicit-explicit strong stability preserving Runge-Kutta methods with high linear order. Strong stability preserving (SSP) time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. The search for high order strong stability preserving time-stepping methods with high order and large allowable time-step has been an active area of research. It is known that implicit SSP Runge-Kutta methods exist only up to sixth order; however, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and we can find implicit SSP Runge-Kutta methods of any linear order. In the current work we find implicit SSP Runge-Kutta methods with high linear order $p_{lin} leq 9$ and nonlinear orders $p=2,3,4$, that are optimal in terms of allowable SSP time-step. Next, we formulate a novel optimization problem for implicit-explicit (IMEX) SSP Runge-Kutta methods and find optimized IMEX SSP Runge-Kutta pairs that have high linear order $p_{lin} leq 7$ and nonlinear orders up to $p=4$. We also find implicit methods with large linear stability regions that pair with known explicit SSP Runge-Kutta methods. These methods are then tested on sample problems to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems.
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References in zbMATH (referenced in 6 articles )
Showing results 1 to 6 of 6.
Sorted by year (- Moradi, A.; Sharifi, M.; Abdi, A.: Transformed implicit-explicit second derivative diagonally implicit multistage integration methods with strong stability preserving explicit part (2020)
- Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal: Strong stability preserving integrating factor two-step Runge-Kutta methods (2019)
- Dimarco, Giacomo; Loubère, Raphaël; Michel-Dansac, Victor; Vignal, Marie-Hélène: Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime (2018)
- Higueras, Inmaculada; Ketcheson, David I.; Kocsis, Tihamér A.: Optimal monotonicity-preserving perturbations of a given Runge-Kutta method (2018)
- Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal: Strong stability preserving integrating factor Runge-Kutta methods (2018)
- Conde, Sidafa; Gottlieb, Sigal; Grant, Zachary J.; Shadid, John N.: Implicit and implicit-explicit strong stability preserving Runge-Kutta methods with high linear order (2017)