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A fourth-order exponential time differencing scheme with real and distinct poles rational approximation for solving

W K Attipoe¹, A Kleefeld^2,3, E O Asante-Asamani¹

¹Clarkson University, 8 Clarkson Ave, Potsdam, 13676, NY, USA.

Journal of Computational and Applied Mathematics

|April 20, 2026

Summary

A new numerical method, ETDRK4RDP, efficiently solves complex reaction-diffusion equations with non-smooth data. This L-stable scheme improves accuracy and parallel performance for scientific computing.

Keywords:

Exponential time differencing fourth-order time stepping non-Padé rational approximation reaction diffusion systems semilinear parabolic problems

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Area of Science:

Numerical analysis
Computational mathematics
Scientific computing

Background:

Reaction-diffusion equations model complex phenomena but are challenging to solve, especially with non-smooth data.
Existing numerical schemes may struggle with accuracy and efficiency for such problems.

Purpose of the Study:

To develop a novel, fourth-order, L-stable numerical scheme for solving non-linear reaction-diffusion systems.
To address challenges posed by non-smooth initial and boundary conditions.

Main Methods:

Developed the ETDRK4RDP scheme, approximating matrix exponentials with a fourth-order, L-acceptable rational function.
Utilized real and distinct poles to avoid complex arithmetic, facilitating parallelization.
Tested the scheme on reaction-diffusion systems with Dirichlet and Neumann boundary conditions.

Main Results:

Empirically verified fourth-order accuracy of the ETDRK4RDP scheme.
Demonstrated efficient damping of spurious oscillations from non-smooth data.
Achieved up to five times speed-up in CPU time compared to other exponential time differencing schemes in parallel implementation.

Conclusions:

The ETDRK4RDP scheme offers an accurate and efficient solution for non-linear reaction-diffusion equations with non-smooth data.
Its design enhances parallelization capabilities, making it attractive for high-performance computing.