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.


References in zbMATH (referenced in 10 articles )

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  1. Izzo, Giuseppe; Jackiewicz, Zdzisław: Strong stability preserving IMEX methods for partitioned systems of differential equations (2021)
  2. Santos, Ricardo; Alves, Leonardo: A comparative analysis of explicit, IMEX and implicit strong stability preserving Runge-Kutta schemes (2021)
  3. Zhang, Hong; Yan, Jingye; Qian, Xu; Gu, Xianming; Song, Songhe: On the preserving of the maximum principle and energy stability of high-order implicit-explicit Runge-Kutta schemes for the space-fractional Allen-Cahn equation (2021)
  4. Zhang, Hong; Yan, Jingye; Qian, Xu; Song, Songhe: Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation (2021)
  5. Moradi, A.; Sharifi, M.; Abdi, A.: Transformed implicit-explicit second derivative diagonally implicit multistage integration methods with strong stability preserving explicit part (2020)
  6. Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal: Strong stability preserving integrating factor two-step Runge-Kutta methods (2019)
  7. 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)
  8. Higueras, Inmaculada; Ketcheson, David I.; Kocsis, Tihamér A.: Optimal monotonicity-preserving perturbations of a given Runge-Kutta method (2018)
  9. Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal: Strong stability preserving integrating factor Runge-Kutta methods (2018)
  10. 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)