1Department of Mathematics, Southern Methodist University | 2Department of Mathematics, Mississippi State University
This repository contains MATLAB files for test problems highlighted in the paper, Vu Thai Luan, Rujeko Chinomona & Daniel R. Reynolds, "Multirate exponential Rosenbrock methods," arXiv:2106.05385, 2021.
Multirate Exponential Rosenbrock (MERB) methods of orders three (MERB3), four (MERB4), five (MERB5),and six (MERB6) are implemented on the additively split multirate problem:
u' = F(t,u) = Jnu + Vnt + Nn(t,u)
where Jn is the Jacobian of F at (tn,un), Vn is the partial derivative of F with respect to time at (tn,un), and Nn(t,u) = F(t,u) - Jnu - Vnt.
MERB methods by design require an update of the Jacobian at each time step, an approach we refer to as dynamic linearization. The fast linear part of the algorithm is Ff(t,u) = Jnu and the slow part is Fs(t,u) = Vnt + Nn(t,u).
We run comparison tests with Multirate Exponential Runge-Kutta (MERK) methods ( third-order MERK3, fourth-order MERK4, and fifth-order MERK5) from V.T. Luan, R. Chinomona and D.R. Reynolds, SIAM Journal on Scientific Computing, 2020 and Multirate Infinitesimal Generalized-structure Additive Runge-Kutta (MRIGARK) methods (third-order MRI-GARK-ERK33a and fourth-order MRI-GARK-ERK45a) from A. Sandu, SIAM Journal on Numerical Analysis, 2019.
MERK and MRIGARK methods use both dynamic linearization and fixed linearization fast-slow splittings. For each test problem optimal time scale separation factors m, which are integer ratios between the slow and fast time step sizes are determined and used to compare methods at their peak efficiency.
The two test problems included in this repository are a 1D reaction-diffusion problem and a non-autonomous bidirectional coupling problem with descriptions in testdescriptions. An analytical solution for the bidirectional coupling problem is provided, while a reference solution for the reaction-diffusion problem is computed by ode15s. The drivers for the considered methods are included in the folders merb, merk, mrigark. For each method and each test problem the drivers output maximum absolute errors, root-mean-square errors, rate of convergence results, number of slow and fast function evaluations, and runtimes.