Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A fourth-order exponential time differencing scheme with dimensional splitting for non-linear reaction-diffusion systems

E. O. Asante-Asamani, A. Kleefeld, B. A. Wade

Abstract

A fourth-order exponential time differencing (ETD) Runge-Kutta scheme with dimensional splitting is developed to solve multidimensional non-linear systems of reaction-diffusion equations (RDE). By approximating the matrix exponential in the scheme with the A-acceptable Padé (2,2) rational function, the resulting scheme (ETDRK4P22-IF) is verified empirically to be fourth-order accurate for several RDE. The scheme is shown to be more efficient than competing fourth-order ETD and IMEX schemes, achieving up to 20 times speed in CPU time. Inclusion of up to three pre-smoothing steps of a lower order L-stable scheme facilitates efficient damping of spurious oscillations arising from problems with non-smooth initial/boundary conditions.

A fourth-order exponential time differencing scheme with dimensional splitting for non-linear reaction-diffusion systems

Abstract

A fourth-order exponential time differencing (ETD) Runge-Kutta scheme with dimensional splitting is developed to solve multidimensional non-linear systems of reaction-diffusion equations (RDE). By approximating the matrix exponential in the scheme with the A-acceptable Padé (2,2) rational function, the resulting scheme (ETDRK4P22-IF) is verified empirically to be fourth-order accurate for several RDE. The scheme is shown to be more efficient than competing fourth-order ETD and IMEX schemes, achieving up to 20 times speed in CPU time. Inclusion of up to three pre-smoothing steps of a lower order L-stable scheme facilitates efficient damping of spurious oscillations arising from problems with non-smooth initial/boundary conditions.
Paper Structure (17 sections, 50 equations, 5 figures, 10 tables)

This paper contains 17 sections, 50 equations, 5 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Materials and Methods
  3. Discretization in space
  4. Discretization in time
  5. Dimensional splitting of fourth-order ETD Runge-Kutta scheme
  6. Padé approximation of the matrix exponential
  7. Partial fraction decomposition of rational functions
  8. Implementation of ETDRK4P22 scheme
  9. Implementation of ETDRK4P22-IF scheme
  10. Results and Discussion
  11. Model problem with Dirichlet boundary
  12. Model problem with Neumann boundary
  13. Enzyme Kinetics
  14. Non-smooth enzyme kinetics with mismatched initial and boundary data
  15. Brusselator in 2D
  16. ...and 2 more sections

Figures (5)

  • Figure 1: Convergence and efficiency plots for Example \ref{['Ex:Model1']}. A) Fourth-order convergence and B) Efficiency of the new fourth-order ETDRK4P22-IF scheme compared with the existing fourth-order ETDRK4P22, second-order ETDRDP-IF and the fourth order SBDF4 schemes.
  • Figure 2: Convergence and efficiency plots for Example \ref{['Ex:Model2']}. A) Exact solution, B) Numerical solution and log-log plots showing C) Fourth-order convergence and D) Efficiency of the new fourth-order ETDRK4P22-IF scheme compared with the existing fourth-order ETDRK4P22, second-order ETDRDP-IF and fourth order SBDF4 schemes.
  • Figure 3: Convergence and efficiency plots for Example \ref{['Ex:Model3']}. Log-log plots showing A) convergence and B) efficiency of ETDRK4P22-IF for the enzyme kinetics problem. Errors are generated between successive numerical solutions with varying temporal step size on a fixed spatial mesh size of $h=0.05$. The diffusion coefficient is $d=0.25$.
  • Figure 4: Numerical solution for Example \ref{['Ex:Model3']} with mismatched initial and boundary data. A) Solution profile using ETDRK4P22-IF with three steps of presmoothing. B) Solution profile using ETDRK4P22-IF without presmoothing. C) Convergence Plots. D) Efficiency Plots. All numerical solutions are generated on a spatial grid with mesh size $h=0.05$. A time step of $k=0.1$ was used for the solutions profiles in A and B. The diffusion coefficient for the model is taken as $d=1$ in all cases.
  • Figure 5: Convergence and efficiency plots for Example \ref{['Ex:Model5']}. A) Numerical solution $U$, B) Numerical solution $V$, log-log plots showing C) Fourth-order convergence, and D) Efficiency of the new fourth-order ETDRK4P22-IF scheme compared with the existing fourth-order ETDRK4P22, second-order ETDRDP-IF and fourth order SBDF4 schemes. Numerical solutions are generated using $40$ nodes per spatial dimension and time step of $0.05$ with final time $T=2$.