RK-stable second derivative multistage methods with strong stability preserving based on Taylor series conditions

Time stepping methods are often required for solving system of ordinary differential equations arising from spatial discretization of partial differential equations. In our prior work, we derived sufficient conditions for a subclass of second derivative general linear methods (SGLMs), so-called second derivative diagonally implicit multistage integration methods, to preserve the strong stability properties of spatial discretizations when coupled with forward Euler and a Taylor series conditions which allows for more flexibility in the choice of spatial discretizations. In this work, we extend this theory to the class of the high number of external stages SGLMs with Runge–Kutta stability. Proposed methods up to order five are examined on some one spatial dimension linear and nonlinear problems in PDEs confirming the verification of the theoretical order and potential of such schemes in maintaining some nonlinear stability properties such as and total variation.