Variable stepsize implicit-explicit general linear methods

Many practical problems in science and engineering are modeled by large systems of ordinary differential equations (ODEs) with additive vector field, whose terms have different stiffness properties. Such a systems can often be written in the form y'(t) = f(y(t))+ g(y(t)), t in [t0, T], y(t0) = y0; y0 in Rm, f: Rm in Rm, g: Rm in Rm, where f(y) represents the non-stiff processes and g(y) represents stiff processes. For efficient integration of this kind of initial value problems we consider implicit-explicit (IMEX) methods, where the non-stiff part f(y) is integrated by an explicit numerical scheme, and the stiff part g(y) is integrated by an implicit numerical scheme. After the investigation of IMEX Runge-Kutta (RK) methods [1], and IMEX General Linear Methods (GLMs) [2,3] in a fixed stepsize formulation, we focus on estimation of local discretization errors and rescaling stepsize techniques for high stage order IMEX GLMs in fixed and variable stepsize environments. We also describe the construction of such methods with desirable accuracy and stability properties. References [1] G. Izzo and Z. Jackiewicz, Highly stable implicit-explicit Runge-Kutta methods, Appl. Numer. Math., Vol. 113, 2017, 71–92. [2] M. Bras, G. Izzo and Z. Jackiewicz, Accurate Implicit–Explicit General Linear Methods with Inherent Runge–Kutta Stability, J. Sci. Comput., Vol. 70(3), 2017, 1105–1143. [3] G. Izzo and Z. Jackiewicz, Transformed implicit-explicit DIMSIMs with strong stability preserving explicit part, Numer. Algorithms, 2019 (in press).