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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=p$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=q=s$$\end{document} and r=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=2$$\end{document} up to order p=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=3$$\end{document} 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.