Analysis and efficient Sylvester-based implementation of a dimension-split ETD2RK scheme for multidimensional reaction–diffusion equations

We propose and analyze a 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 involving the exact matrix exponential. To enable efficient implementation, we employ Padé 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 Padé-based ETD2RK-DS formulation is provided, yielding a first-order error bound. 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 O(m³) to O(m²) in two dimensions and from O(m⁵) to O(m³) 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, demonstrate the stability and computational efficiency of the proposed framework, as well as the substantial computational advantages of the Sylvester-based implementation over classical LU-based solvers.