Application of approximate matrix factorization to high order linearly implicit Runge-Kutta methods

Linearly implicit Runge-Kutta methods with approximate matrix factorization can solve efficiently large systems of differential equations that have a stiff linear part, e.g. reaction-diffusion systems. However, the use of approximate factorization usually leads to loss of accuracy, which makes it attractive only for low order time integration schemes. This paper discusses the application of approximate matrix factorization with high order methods; an inexpensive correction procedure applied to each stage allows to retain the high order of the underlying linearly implicit Runge-Kutta scheme. The accuracy and stability of the methods are studied. Numerical experiments on reaction-diffusion type problems of different sizes and with different degrees of stiffness illustrate the efficiency of the proposed approach

A Study of the Numerical Stability of an ImEx Scheme with Application to the Poisson-Nernst-Planck Equations

The Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions (PNP-FBV) describe ion transport with Faradaic reactions and have applications in a wide variety of fields. Using an adaptive time-stepper based on a second-order variable step-size, semi-implicit, backward differentiation formula (VSSBDF2), we observe that when the underlying dynamics is one that would have the solutions converge to a steady state solution, the adaptive time-stepper produces solutions that "nearly" converge to the steady state and that, simultaneously, the time-step sizes stabilize at a limiting size $dt_\infty$. Linearizing the SBDF2 scheme about the steady state solution, we demonstrate that the linearized scheme is conditionally stable and that this is the cause of the adaptive time-stepper's behaviour. Mesh-refinement, as well as a study of the eigenvectors corresponding to the critical eigenvalues, demonstrate that the conditional stability is not due to a time-step constraint caused by high-frequency contributions. We study the stability domain of the linearized scheme and find that it can have corners as well as jump discontinuities.Comment: The earlier version, arXiv:1905.01368v1, also contained: a linear stability analysis of the SBDF2 scheme, a study of the effect of Richardson Extrapolation on numerical stability, and a study of the stability domain of the logistic equation. This is the 2nd of a pair of articles: "A Variable Step Size Implicit-Explicit Scheme for the Solution of the Poisson-Nernst-Planck Equations", is on arXi