On Starting Procedures for IMEX General Linear Methods based on Generalized Runge-Kutta Schemes

Abstract

General linear methods (GLMs) are an efficient class of time integration schemes to solve time-dependent differential equations, and GLMs need an initial input vector to start. Starting procedures are dedicated numerical schemes used to obtain the input vectors for executing GLMs. In this article, we present the construction of some starting procedures for implicit-explicit (IMEX) GLM solvers that are specially designed for partitioned differential systems. The construction of starting procedures is based on the generalized Runge-Kutta schemes in an IMEX environment, and we generate a sequence of starting procedures for computing the components of the initial input vector. We demonstrate examples of starting procedures of orders, $p=2$ and $p=3$. We provide two numerical illustrations of a time-dependent partitioned differential system, where IMEX-GLM is employed in conjunction with the proposed procedures. The reported numerical results confirm the desired convergence and do not report any order reduction.