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

Ibrahim O. Sarumi

TL;DR

This work develops a robust, second-order, dimension-splitting exponential time differencing scheme (ETD2RK-DS) for multidimensional reaction–diffusion equations, combining a Padé-based fully discrete formulation with a tensor-slicing strategy that reduces multidimensional solves to independent 1D problems. It provides rigorous stability and convergence analysis, including explicit Padé-error bounds, and introduces a Sylvester-equation reformulation to enable a spectral, reusable eigensolution pipeline that substantially lowers per-step cost relative to LU-based solvers, especially for higher Padé degrees and heterogeneous diffusivities. The authors validate the method on 2D and 3D problems (e.g., Allen–Cahn and FitzHugh–Nagumo), confirming second-order temporal accuracy and scalable performance, with numerical results showing strong agreement between LU and Sylvester implementations. Overall, the approach delivers a practical, scalable framework for efficient high-resolution simulations of reaction–diffusion systems in multiple dimensions, combining theoretical guarantees with implementable, tensor-based algorithms.

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 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. 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.

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

TL;DR

This work develops a robust, second-order, dimension-splitting exponential time differencing scheme (ETD2RK-DS) for multidimensional reaction–diffusion equations, combining a Padé-based fully discrete formulation with a tensor-slicing strategy that reduces multidimensional solves to independent 1D problems. It provides rigorous stability and convergence analysis, including explicit Padé-error bounds, and introduces a Sylvester-equation reformulation to enable a spectral, reusable eigensolution pipeline that substantially lowers per-step cost relative to LU-based solvers, especially for higher Padé degrees and heterogeneous diffusivities. The authors validate the method on 2D and 3D problems (e.g., Allen–Cahn and FitzHugh–Nagumo), confirming second-order temporal accuracy and scalable performance, with numerical results showing strong agreement between LU and Sylvester implementations. Overall, the approach delivers a practical, scalable framework for efficient high-resolution simulations of reaction–diffusion systems in multiple dimensions, combining theoretical guarantees with implementable, tensor-based algorithms.

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 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. 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 to in two dimensions and from to in three dimensions when compared with banded LU solvers for the unsplit problem, where 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.
Paper Structure (17 sections, 7 theorems, 116 equations, 3 figures, 7 tables, 2 algorithms)

This paper contains 17 sections, 7 theorems, 116 equations, 3 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Numerical method
  3. Spatial Discretization
  4. Time-stepping Scheme
  5. Numerical Implementation
  6. Padé-based ETD2RK--DS formulation
  7. Dimension Splitting and Problem-Size Reduction
  8. Two-Dimensional Case
  9. Three-Dimension Case
  10. Efficient Linear Solvers
  11. LU Factorization-Based Implementation
  12. Sylvester Equation-Based Implementation
  13. Algorithmic Realization
  14. Stability and Convergence Analysis
  15. Padé Approximation Error
  16. ...and 2 more sections

Key Result

Proposition 2.1

Suppose $A$ and $B$ are square matrices with real entries such that $AB = BA$. Then the following relations hold:

Figures (3)

  • Figure 1: Example \ref{['exp:Allen-Cahn']}, comparison of the pointwise relative approximation error, for different values of $N$, 2D (left), 3D (right)
  • Figure 2: Example \ref{['exp:singular']}: numerical solution profiles for the two-dimensional problem at $y=0.5$. Short-time evolution ($0\le t\le1$, left) and long-time evolution ($0\le t\le100$, right)
  • Figure 3: Example \ref{['exp:FHN']} 3D plot of the solution components $u$ (left) and $v$ (right) at $t = 0.125$, taking $N = 256$

Theorems & Definitions (19)

  • Proposition 2.1
  • Remark 1
  • Theorem 4.2: Stability
  • proof
  • Remark 2
  • Proposition 4.3
  • proof
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • ...and 9 more