Strong stability preserving implicit and implicit–explicit second derivative general linear methods with RK stability

In this work, we use a formulation based on forward Euler and backward derivative condition to obtain A -stable SSP implicit SGLMs up to order five and stage order q=p and SSP implicit–explicit (IMEX) SGLMs where the implicit part of the method is A -stable and the time-step is apart from the explicit part. These kind of methods compared to explicit ones of the same order and number of stages have quite larger SSP time-step. Moreover, the construction of SSP IMEX schemes of order p=q=s and r=2 up to order p=3 is presented where the implicit part of the method has Runge–Kutta stability together with A -stability property. Numerical results to show the expected order of convergence of the proposed methods are presented on a variety of linear and nonlinear problems.