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.