Related papers: Strong-stability-preserving additive linear multis…
Strong stability preserving (SSP) methods are designed primarily for time integration of nonlinear hyperbolic PDEs, for which the permissible SSP step size varies from one step to the next. We develop the first SSP linear multistep methods…
High-order spatial discretizations with strong stability properties (such as monotonicity) are desirable for the solution of hyperbolic PDEs. Methods may be compared in terms of the strong stability preserving (SSP) time-step. We prove an…
Strong stability preserving (SSP) integrators for initial value ODEs preserve temporal monotonicity solution properties in arbitrary norms. All existing SSP methods, including implicit methods, either require small step sizes or achieve…
When evolving in time the solution of a hyperbolic partial differential equation, it is often desirable to use high order strong stability preserving (SSP) time discretizations. These time discretizations preserve the monotonicity…
Strong stability preserving (SSP) coefficients govern the maximally allowable step-size at which positivity or contractivity preservation of integration methods for initial value problems is guaranteed. In this paper, we show that the task…
This paper extends deterministic notions of Strong Stability Preservation (SSP) to the stochastic setting, enabling nonlinearly stable numerical solutions to stochastic differential equations (SDEs) and stochastic partial differential…
High order strong stability preserving (SSP) time discretizations are often needed to ensure the nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed for the solution of…
We prove the existence of explicit linear multistep methods of any order with positive coefficients. Our approach is based on formulating a linear programming problem and establishing infeasibility of the dual problem. This yields a number…
High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The…
Linear multistep methods (LMMs) applied to approximate the solution of initial value problems---typically arising from method-of-lines semidiscretizations of partial differential equations---are often required to have certain monotonicity…
High order strong stability preserving (SSP) time discretizations are advantageous for use with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. The search for high order strong stability…
Strong Stability Preserving (SSP) time integration schemes maintain stability of the forward Euler method for any initial value problem. However, only a small subset of Runge-Kutta (RK) methods are SSP, and many efficient high-order time…
We present a theoretical analysis of stochastic optimization methods in terms of their sensitivity with respect to the step size. We identify a key quantity that, for each method, describes how the performance degrades as the step size…
High order strong stability preserving (SSP) time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs. Over the past decade…
We construct a family of embedded pairs for optimal strong stability preserving explicit Runge-Kutta methods of order $2 \leq p \leq 4$ to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction,…
Strong stability preserving (SSP) Runge-Kutta methods are often desired when evolving in time problems that have two components that have very different time scales. Where the SSP property is needed, it has been shown that implicit and…
Problems that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, where…
In this paper, nonstandard multistep methods are considered. It is shown that under some (sufficient and necessary) conditions, these methods attain the same order as their standard counterparts - to prove this statement, a nonstandard…
We report on recent work on adaptive timestep control for weakly instationary gas flows [16, 18, 17] carried out within SFB 401, TPA3. The method which we implement and extend is a space-time splitting of adjoint error representations for…
In this paper we continue our work on adaptive timestep control for weakly non- stationary problems. The core of the method is a space-time splitting of adjoint error representations for target functionals due to S\"uli and Hartmann. The…