High-order explicit second derivative methods with strong stability properties based on Taylor series conditions
Keywords:Second derivative methods, monotonicity, strong stability preserving, Taylor series, Euler equation
When faced with the task of solving hyperbolic partial differential equations (PDEs), high order, strong stability-preserving (SSP) time integration methods are often needed to ensure preservation of the nonlinear strong stability properties of spatial discretizations. Among such methods, SSP second derivative time-stepping schemes have been recently introduced and used for evolving hyperbolic PDEs. In previous works, coupling of forward Euler and a second derivative formulation led to sufficient conditions for a second derivative general linear method (SGLM), which preserve the strong stability properties of spatial discretizations. However, for such methods, the types of spatial discretizations that can be used are limited. In this paper, we use a formulation based on forward Euler and Taylor series conditions to extend the SSP SGLM framework. We investigate the construction of SSP second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of SGLMs with order \(p=r=s\) and stage order \(q=p, p−1\) up to order eight, where r is the number of external stages and s is the number of internal stages of the method. Proposed methods are examined on some one-dimensional linear and nonlinear systems to verify their theoretical order, and show potential of these schemes in preserving some nonlinear stability properties such as positivity and total variation.