# 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.