Analysis and Efficient Sylvester-Based Implementation of a Dimension-Split ETD2RK Scheme for Multidimensional Reaction-Diffusion Equations

Abstract

We propose and analyze a second-order, dimension-split exponential time differencing Runge--Kutta scheme (ETD2RK-DS) for multidimensional reaction--diffusion equations in two and three spatial dimensions. Under mild assumptions on the nonlinear source term, we establish uniform stability bounds and prove second-order temporal convergence for the underlying dimension-split scheme. To enable efficient implementation, we employ Pad\'e approximations of the matrix exponential, converting each required matrix-exponential--vector product into the solution of a shifted linear system. A convergence analysis of the resulting Pad\'e-based ETD2RK-DS formulation is provided. We derive explicit and reproducible tensor-slicing and reshaping algorithms that realize the dimension-splitting strategy, decomposing multidimensional systems into collections of independent one-dimensional problems. This leads to a reduction of the dominant per-time-step computational cost from $\mathcal{O}(m^3)$ to $\mathcal{O}(m^2)$ in two dimensions and from $\mathcal{O}(m^5)$ to $\mathcal{O}(m^3)$ in three dimensions when compared with banded LU solvers for the unsplit problem, where $m$ denotes the number of grid points per spatial direction. Furthermore, we develop a Sylvester-equation reformulation of the resulting one-dimensional systems, enabling a highly efficient spectral implementation based on reusable eigendecompositions, matrix--vector multiplications, and Hadamard divisions. Numerical experiments in two and three dimensions, including a coupled FitzHugh--Nagumo system, confirm the second-order temporal accuracy, stability of the underlying scheme, and scalability of the proposed ETD2RK-DS framework, as well as the substantial computational advantages of the Sylvester-based implementation over classical LU-based solvers.