# American Institute of Mathematical Sciences

December  2019, 24(12): 6387-6417. doi: 10.3934/dcdsb.2019144

## A hybrid method for stiff reaction–diffusion equations

 1 Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA 2 Department of Mathematics, University of California, Riverside, Riverside, CA 92507, USA 3 Department of Mathematics, Department of Developmental and Cell Biology, University of California, Irvine, Irvine, CA 92697, USA

* Corresponding author: Qing Nie

Received  October 2018 Revised  January 2019 Published  July 2019

The second-order implicit integration factor method (IIF2) is effective at solving stiff reaction–diffusion equations owing to its nice stability condition. IIF has previously been applied primarily to systems in which the reaction contained no explicitly time-dependent terms and the boundary conditions were homogeneous. If applied to a system with explicitly time-dependent reaction terms, we find that IIF2 requires prohibitively small time-steps, that are relative to the square of spatial grid sizes, to attain its theoretical second-order temporal accuracy. Although the second-order implicit exponential time differencing (iETD2) method can accurately handle explicitly time-dependent reactions, it is more computationally expensive than IIF2. In this paper, we develop a hybrid approach that combines the advantages of both methods, applying IIF2 to reaction terms that are not explicitly time-dependent and applying iETD2 to those which are. The second-order $\underline {\text{h}} {\text{ybrid}}$ ${\text{I}}\underline {{\text{IF}}} - \underline {\text{E}} {\text{TD}}$ method (hIFE2) inherits the lower complexity of IIF2 and the ability to remain second-order accurate in time for large time-steps from iETD2. Also, it inherits the unconditional stability from IIF2 and iETD2 methods for dealing with the stiffness in reaction–diffusion systems. Through a transformation, hIFE2 can handle nonhomogeneous boundary conditions accurately and efficiently. In addition, this approach can be naturally combined with the compact and array representations of IIF and ETD for systems in higher spatial dimensions. Various numerical simulations containing linear and nonlinear reactions are presented to demonstrate the superior stability, accuracy, and efficiency of the new hIFE method.

Citation: Yuchi Qiu, Weitao Chen, Qing Nie. A hybrid method for stiff reaction–diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6387-6417. doi: 10.3934/dcdsb.2019144
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##### References:
Plots of the numerical error at $T = 1$ after applying IIF2, iETD2, and hIFE2 to the scalar equation in (8) with $u(0) = 1$ for various $\Delta t$. Plots are shown for (A) $f(u,t) = t^2$ with $\alpha = -10^1,\ -10^2,\ -10^3,\ -10^4,\ -10^5$, and $-10^6$; (B) $f(u,t) = -u$ with $\alpha = -8,\ -16,\ -32,\ -64$, and $-128$; and (C) $f(u,t) = -u+t^2$ with $\alpha = -10^2,\ -10^3,\ -10^4,\ -10^5$, and $-10^6$. The curves for iETD2 and hIFE2 are identical in (A), and those for IIF2 and hIFE2 are identical in (B). We see that for the time-dependent reactions (A, C), the error in IIF2 increases as $-\alpha$ increases while the error in iETD2 and hIFE2 decreases
The temporal errors at $T = 1$ in the maximum norm when solving the semi-discrete form (16) of (27) for different reactions with the IIF, iETD2, and hIFE2 methods. In all simulations, the reaction coefficient $d = 1$. (A) IIF2 for $F(U,t) = t^2$; (B) iETD2 for $F(U,t) = t^2$; (C) hIFE2 for $F(U,t) = t^2$; (D) IIF2 for $F(U,t) = -U$; (E) iETD2 for $F(U,t) = -U$; (F) hIFE2 for $F(U,t) = -U$; (G) IIF2 for $F(U,t) = -U+t^2$; (H) iETD2 for $F(U,t) = -U+t^2$; (I) hIFE2 for $F(U,t) = -U+t^2$. Different colors represent the number of points, $N$, in the spatial discretization, where $N = 32,\ 64,\ 128,\ 256,\ 512$, and $1024$. Subfigures in same row share the same $y$-axis while subfigures in same column share the same $x$-axis. Panels (B) and (C) are identical because hIFE2 treats time-dependent terms with iETD2, and panels (D) and (F) are identical since hIFE2 treats autonomous terms with IIF2
Plots of the numerical error at $T = 1$ in maximum norm after applying hIFE2 to (27) with Neumann, Dirichlet, and mixed boundary conditions for various $\Delta t$ and fixed $N$. The hIFE2 is applied to both original and transformed (Section 3.2) equations. Plots are shown for hIFE2 on: (A) the original equation with Neumann boundary; (B) the original equation with Dirichlet boundary; (C) the original equation with mixed boundary; (D) the transformed equation with Neumann boundary; (E) the transformed equation with Dirichlet boundary; (F) the transformed equation with mixed boundary. Different colors represent different spatial mesh sizes $N$, where $N = 32,\ 64,\ 128,\ 256,\ 512$, and $1024$
The truncation errors of IIF2, iETD2, and hIFE2 when applied to (8) with different reactions
 Method Reaction $f$ Truncation error IIF2 $g(t)$ $-\frac{1}{12}\Delta t^2(\alpha ^2g_n-2\alpha g'_n+g''_n)$ $-\frac{1}{24}\Delta t^3(\alpha ^3g_n-\alpha ^2 g'_n-\alpha g''_n+g'''_n)+\mathcal{O}(\Delta t^4)$ $ru$ $-\frac{1}{12}\Delta t^2r^3u_n$ $-\frac{1}{24}\Delta t^3(2\alpha r^3+r^4)u_n+\mathcal{O}(\Delta t^4)$ $ru+g(t)$ $-\frac{1}{12}\Delta t^2 \big[ \alpha^2g_n+\alpha(-rg_n-2g_n')+(r^3u_n+r^2g_n+rg_n'+g_n'') \big]$ $-\frac{1}{24}\Delta t^3\big[ \alpha^3g_n+\alpha^2(-rg_n-g_n')+\alpha(2r^3u_n+r^2g_n-g_n'')$ $+(r^4u_n+r^3g_n+r^2g_n'+rg_n''+g_n''')\big]+\mathcal{O}(\Delta t^4)$ iETD2 $g(t)$ $-\frac{1}{12}\Delta t^2g''_n$ $-\frac{1}{24}\Delta t^3(\alpha g''_n+g'''_n)+\mathcal{O}(\Delta t^4)$ $ru$ $-\frac{1}{12}\Delta t^2(\alpha^2r+2\alpha r^2+r^3)u_n$ $-\frac{1}{24}\Delta t^3(2\alpha^3 r+5\alpha^2 r^2+4\alpha r^3+r^4)u_n+\mathcal{O}(\Delta t^4)$ $ru+g(t)$ $-\frac{1}{12}\Delta t^2 \big[\alpha^2ru_n+\alpha(2r^2u_n+rg_n)+(r^3u_n+r^2g_n+rg_n'+g_n'')\big]$ $-\frac{1}{24}\Delta t^3\big[2\alpha^3ru_n-\alpha^2(5r^2u_n+2rg_n)+\alpha(4r^3u_n+3r^2g_n+2rg_n'+g_n'')$ $+(r^4u_n+r^3g_n+r^2g_n'+rg_n''+g_n''')\big]+\mathcal{O}(\Delta t^4)$ hIFE2 $g(t)$ equivalent to iETD2 $ru$ equivalent to IIF2 $ru+g(t)$ $-\frac{1}{12}\Delta t^2 \big[-\alpha rg_n+(r^3u_n+r^2g_n+rg_n'+g_n'')\big]$ $-\frac{1}{24}\Delta t^3\big[-\alpha^2rg_n+\alpha(2r^3u_n+r^2g_n+g_n'')$ $+(r^4u_n+r^3g_n+r^2g_n'+rg_n''+g_n''')\big]+\mathcal{O}(\Delta t^4)$
 Method Reaction $f$ Truncation error IIF2 $g(t)$ $-\frac{1}{12}\Delta t^2(\alpha ^2g_n-2\alpha g'_n+g''_n)$ $-\frac{1}{24}\Delta t^3(\alpha ^3g_n-\alpha ^2 g'_n-\alpha g''_n+g'''_n)+\mathcal{O}(\Delta t^4)$ $ru$ $-\frac{1}{12}\Delta t^2r^3u_n$ $-\frac{1}{24}\Delta t^3(2\alpha r^3+r^4)u_n+\mathcal{O}(\Delta t^4)$ $ru+g(t)$ $-\frac{1}{12}\Delta t^2 \big[ \alpha^2g_n+\alpha(-rg_n-2g_n')+(r^3u_n+r^2g_n+rg_n'+g_n'') \big]$ $-\frac{1}{24}\Delta t^3\big[ \alpha^3g_n+\alpha^2(-rg_n-g_n')+\alpha(2r^3u_n+r^2g_n-g_n'')$ $+(r^4u_n+r^3g_n+r^2g_n'+rg_n''+g_n''')\big]+\mathcal{O}(\Delta t^4)$ iETD2 $g(t)$ $-\frac{1}{12}\Delta t^2g''_n$ $-\frac{1}{24}\Delta t^3(\alpha g''_n+g'''_n)+\mathcal{O}(\Delta t^4)$ $ru$ $-\frac{1}{12}\Delta t^2(\alpha^2r+2\alpha r^2+r^3)u_n$ $-\frac{1}{24}\Delta t^3(2\alpha^3 r+5\alpha^2 r^2+4\alpha r^3+r^4)u_n+\mathcal{O}(\Delta t^4)$ $ru+g(t)$ $-\frac{1}{12}\Delta t^2 \big[\alpha^2ru_n+\alpha(2r^2u_n+rg_n)+(r^3u_n+r^2g_n+rg_n'+g_n'')\big]$ $-\frac{1}{24}\Delta t^3\big[2\alpha^3ru_n-\alpha^2(5r^2u_n+2rg_n)+\alpha(4r^3u_n+3r^2g_n+2rg_n'+g_n'')$ $+(r^4u_n+r^3g_n+r^2g_n'+rg_n''+g_n''')\big]+\mathcal{O}(\Delta t^4)$ hIFE2 $g(t)$ equivalent to iETD2 $ru$ equivalent to IIF2 $ru+g(t)$ $-\frac{1}{12}\Delta t^2 \big[-\alpha rg_n+(r^3u_n+r^2g_n+rg_n'+g_n'')\big]$ $-\frac{1}{24}\Delta t^3\big[-\alpha^2rg_n+\alpha(2r^3u_n+r^2g_n+g_n'')$ $+(r^4u_n+r^3g_n+r^2g_n'+rg_n''+g_n''')\big]+\mathcal{O}(\Delta t^4)$
Eigenvalues of $A$, $\lambda_j$, under different spatial resolutions, where $d = 1$, $a = 0$, $b = \pi/2$, $j = 1,\ 5,\ N/2,\ N$
 32 64 128 256 512 1024 1 $-$1.00 $-$1.00 $-$1.00 $-$1.00 $-$1.00 $-$1.00 5 $-$7.97e+1 $-$8.07e+1 $-$8.09e+1 $-$8.09e+1 $-$8.10e+1 $-$8.10e+1 $N/2$ $-$7.89e+2 $-$3.23e+03 $-$1.31e+04 $-$5.28e+04 $-$2.12e+05 $-$8.49e+05 $N$ $-$1.66e+03 $-$6.64e+03 $-$2.66e+04 $-$1.06e+05 $-$4.25e+05 $-$1.70e+06
 32 64 128 256 512 1024 1 $-$1.00 $-$1.00 $-$1.00 $-$1.00 $-$1.00 $-$1.00 5 $-$7.97e+1 $-$8.07e+1 $-$8.09e+1 $-$8.09e+1 $-$8.10e+1 $-$8.10e+1 $N/2$ $-$7.89e+2 $-$3.23e+03 $-$1.31e+04 $-$5.28e+04 $-$2.12e+05 $-$8.49e+05 $N$ $-$1.66e+03 $-$6.64e+03 $-$2.66e+04 $-$1.06e+05 $-$4.25e+05 $-$1.70e+06
Different boundary conditions in (27), and their corresponding $A$ and $B(t)$ in the semi-discrete form (26)
 Neumann Dirichlet Mixed BCs $u_x\vert_{x=0}=e^{-2t}\cos \frac{\pi}{6}$ $u\vert_{x=0}=e^{-2t}\sin \frac{\pi}{6},$ $u_x\vert_{x=0}=e^{-2t}\cos \frac{\pi}{6}$ $u_x\vert_{x=\frac{\pi}{2}}=e^{-2t}\cos \frac{2\pi}{3}$ $u\vert_{x=\frac{\pi}{2}}=e^{-2t}\sin \frac{2\pi}{3}$ $u\vert_{x=\frac{\pi}{2}}=e^{-2t}\sin \frac{2\pi}{3}$ $B(t)$ $e^{-2t}{ \begin{bmatrix}-\frac{2\cos \frac{\pi}{6}}{\Delta x}\\0\\ \vdots \\0 \\\frac{2\cos \frac{2\pi}{3}}{\Delta x}\end{bmatrix}_{N+1}}$ $e^{-2t}{\begin{bmatrix}\frac{\sin \frac{\pi}{6}}{\Delta x^2}\\ 0\\ \vdots \\0 \\ \frac{\sin \frac{2\pi}{3}}{\Delta x^2}\end{bmatrix}_{N-1}}$ $e^{-2t}{\begin{bmatrix}-\frac{2\cos \frac{\pi}{6}}{\Delta x} \\0\\ \vdots \\0\\ \frac{\sin \frac{2\pi}{3}}{\Delta x^2}\end{bmatrix}_N}$ $A$ $\frac{1}{\Delta x^2} \begin{bmatrix} -2&2&&\\ 1&-2&1\\ &\ddots&\ddots&\ddots\\ &&1&-2&1\\ &&&2&-2\\ \end{bmatrix}_{(N+1)^2}$ $\frac{1}{\Delta x^2} \begin{bmatrix} -2&1&&\\ 1&-2&1\\ &\ddots&\ddots&\ddots\\ &&1&-2&1\\ &&&1&-2\\ \end{bmatrix}_{(N-1)^2}$ $\frac{1}{\Delta x^2} \begin{bmatrix} -2&2&&\\ 1&-2&1\\ &\ddots&\ddots&\ddots\\ &&1&-2&1\\ &&&1&-2\\ \end{bmatrix}_{N^2}$
 Neumann Dirichlet Mixed BCs $u_x\vert_{x=0}=e^{-2t}\cos \frac{\pi}{6}$ $u\vert_{x=0}=e^{-2t}\sin \frac{\pi}{6},$ $u_x\vert_{x=0}=e^{-2t}\cos \frac{\pi}{6}$ $u_x\vert_{x=\frac{\pi}{2}}=e^{-2t}\cos \frac{2\pi}{3}$ $u\vert_{x=\frac{\pi}{2}}=e^{-2t}\sin \frac{2\pi}{3}$ $u\vert_{x=\frac{\pi}{2}}=e^{-2t}\sin \frac{2\pi}{3}$ $B(t)$ $e^{-2t}{ \begin{bmatrix}-\frac{2\cos \frac{\pi}{6}}{\Delta x}\\0\\ \vdots \\0 \\\frac{2\cos \frac{2\pi}{3}}{\Delta x}\end{bmatrix}_{N+1}}$ $e^{-2t}{\begin{bmatrix}\frac{\sin \frac{\pi}{6}}{\Delta x^2}\\ 0\\ \vdots \\0 \\ \frac{\sin \frac{2\pi}{3}}{\Delta x^2}\end{bmatrix}_{N-1}}$ $e^{-2t}{\begin{bmatrix}-\frac{2\cos \frac{\pi}{6}}{\Delta x} \\0\\ \vdots \\0\\ \frac{\sin \frac{2\pi}{3}}{\Delta x^2}\end{bmatrix}_N}$ $A$ $\frac{1}{\Delta x^2} \begin{bmatrix} -2&2&&\\ 1&-2&1\\ &\ddots&\ddots&\ddots\\ &&1&-2&1\\ &&&2&-2\\ \end{bmatrix}_{(N+1)^2}$ $\frac{1}{\Delta x^2} \begin{bmatrix} -2&1&&\\ 1&-2&1\\ &\ddots&\ddots&\ddots\\ &&1&-2&1\\ &&&1&-2\\ \end{bmatrix}_{(N-1)^2}$ $\frac{1}{\Delta x^2} \begin{bmatrix} -2&2&&\\ 1&-2&1\\ &\ddots&\ddots&\ddots\\ &&1&-2&1\\ &&&1&-2\\ \end{bmatrix}_{N^2}$
Numerical errors in terms of the maximum norm and CPU time for the various methods on the example in Section 5.1 at $T = 1$ with diffusion coefficient $d = 2$. Here $N$ is the number of grid points in the spatial discretization ($\Delta x = \pi/2N$), and the time step $\Delta t = 0.1\Delta x$. "CPU time 1" is the CPU time for initializing the matrices (Appendix C), "CPU time 2" is the CPU time for the iterations, and "CPU time" is the sum of the two
 $N$ $L^{\infty}$ error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) IIF2 8 0.00228 - 0.09 0.05 0.04 16 0.000591 1.95 0.04 0.02 0.02 32 0.000198 1.58 0.07 0.03 0.04 64 7.81e-05 1.34 0.13 0.04 0.09 128 0.000108 $-$0.46 0.54 0.07 0.47 256 5.18e-05 1.06 1.26 0.23 1.03 512 1.83e-05 1.50 4.00 1.39 2.61 1024 2.07e-05 $-$0.18 28.30 7.75 20.55 2048 1.07e-05 0.96 168.12 42.10 126.02 4096 5.35e-06 1.00 1148.42 265.35 883.07 $N$ $L^{\infty}$ error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) iETD2 8 0.00216 - 0.07 0.04 0.03 16 0.000539 2.00 0.07 0.04 0.03 32 0.000135 2.00 0.12 0.06 0.06 64 3.37e-05 2.00 0.80 0.07 0.73 128 8.41e-06 2.00 3.78 0.16 3.62 256 2.1e-06 2.00 22.99 0.54 22.45 512 5.26e-07 2.00 289.66 2.70 286.96 1024 1.32e-07 2.00 2841.66 14.65 2827.01 2048 3.31e-08 1.99 35348.32 91.84 35256.48 4096 - - $\text{too long}$ - - $N$ $L^{\infty}$ error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) hIFE2 8 0.00217 - 0.12 0.09 0.03 16 0.000544 1.99 0.06 0.04 0.02 32 0.000137 1.99 0.08 0.05 0.03 64 3.42e-05 2.00 0.16 0.08 0.08 128 8.75e-06 1.97 0.76 0.17 0.59 256 2.21e-06 1.99 1.85 0.54 1.31 512 5.53e-07 2.00 9.17 2.61 6.56 1024 1.49e-07 1.89 61.82 14.20 47.62 2048 3.93e-08 1.93 419.24 89.49 329.75 4096 1.12e-08 1.81 3096.23 603.04 2493.19 $N$ $L^{\infty}$ error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) fEIF2 8 0.00216 - 0.37 0.37 0.00 16 0.00054 2.00 0.04 0.04 0.00 32 0.000135 2.00 0.07 0.07 0.00 64 3.38e-05 2.00 0.09 0.08 0.01 128 8.44e-06 2.00 0.54 0.18 0.36 256 2.11e-06 2.00 1.41 0.69 0.72 512 5.28e-07 2.00 11.62 3.01 8.61 1024 1.32e-07 2.00 84.11 16.11 68.00 2048 3.31e-08 1.99 613.91 101.12 512.79 4096 8.89e-09 1.90 4700.11 707.64 3992.47
 $N$ $L^{\infty}$ error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) IIF2 8 0.00228 - 0.09 0.05 0.04 16 0.000591 1.95 0.04 0.02 0.02 32 0.000198 1.58 0.07 0.03 0.04 64 7.81e-05 1.34 0.13 0.04 0.09 128 0.000108 $-$0.46 0.54 0.07 0.47 256 5.18e-05 1.06 1.26 0.23 1.03 512 1.83e-05 1.50 4.00 1.39 2.61 1024 2.07e-05 $-$0.18 28.30 7.75 20.55 2048 1.07e-05 0.96 168.12 42.10 126.02 4096 5.35e-06 1.00 1148.42 265.35 883.07 $N$ $L^{\infty}$ error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) iETD2 8 0.00216 - 0.07 0.04 0.03 16 0.000539 2.00 0.07 0.04 0.03 32 0.000135 2.00 0.12 0.06 0.06 64 3.37e-05 2.00 0.80 0.07 0.73 128 8.41e-06 2.00 3.78 0.16 3.62 256 2.1e-06 2.00 22.99 0.54 22.45 512 5.26e-07 2.00 289.66 2.70 286.96 1024 1.32e-07 2.00 2841.66 14.65 2827.01 2048 3.31e-08 1.99 35348.32 91.84 35256.48 4096 - - $\text{too long}$ - - $N$ $L^{\infty}$ error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) hIFE2 8 0.00217 - 0.12 0.09 0.03 16 0.000544 1.99 0.06 0.04 0.02 32 0.000137 1.99 0.08 0.05 0.03 64 3.42e-05 2.00 0.16 0.08 0.08 128 8.75e-06 1.97 0.76 0.17 0.59 256 2.21e-06 1.99 1.85 0.54 1.31 512 5.53e-07 2.00 9.17 2.61 6.56 1024 1.49e-07 1.89 61.82 14.20 47.62 2048 3.93e-08 1.93 419.24 89.49 329.75 4096 1.12e-08 1.81 3096.23 603.04 2493.19 $N$ $L^{\infty}$ error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) fEIF2 8 0.00216 - 0.37 0.37 0.00 16 0.00054 2.00 0.04 0.04 0.00 32 0.000135 2.00 0.07 0.07 0.00 64 3.38e-05 2.00 0.09 0.08 0.01 128 8.44e-06 2.00 0.54 0.18 0.36 256 2.11e-06 2.00 1.41 0.69 0.72 512 5.28e-07 2.00 11.62 3.01 8.61 1024 1.32e-07 2.00 84.11 16.11 68.00 2048 3.31e-08 1.99 613.91 101.12 512.79 4096 8.89e-09 1.90 4700.11 707.64 3992.47
Numerical errors and CPU time for the test in Section 5.2 at time $T = 1$. We set the diffusion coefficient $d = 0.1$ and the coefficients of the reactions $a = 500$ and $b = -2$. For each simulation, we fix the number of grid points $N = 1024$ ($\Delta x = \pi/2N$), and run the simulation for $K$ time steps ($\Delta t = T/K$). The error $e$ is measured in the maximum norm, and the relative error is defined by $e / \max\{ \| U_K\|_{\infty},\| V_K\|_{\infty} \}$, where $U_K$ and $V_K$ are the numerical solutions after $K$ time steps. "CPU time 1" is the CPU time for initialization (Appendix C), "CPU time 2" is the CPU time for the iterations, and "CPU time" is the sum of the two
 $K$ $L^{\infty}$ error Relative error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) IIF2 20 10 0.00381 - 5.32 5.25 0.07 40 4.81 0.00182 1.06 5.07 4.91 0.16 80 2.32 0.000881 1.05 5.09 4.78 0.31 160 1.12 0.000425 1.05 5.07 4.44 0.63 320 0.534 0.000203 1.07 5.24 3.90 1.34 640 0.251 9.51e-05 1.09 5.92 3.40 2.52 1280 0.115 4.34e-05 1.13 7.90 2.92 4.98 2560 0.0503 1.91e-05 1.19 12.84 2.55 10.29 $K$ $L^{\infty}$ error Relative error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) iETD2 20 3.99 0.00151 - 19.63 10.88 8.75 40 0.994 0.000377 2.00 28.60 10.80 17.80 80 0.248 9.41e-05 2.00 46.92 10.76 36.16 160 0.0617 2.34e-05 2.01 80.10 10.41 69.69 320 0.0152 5.76e-06 2.02 148.60 9.80 138.80 640 0.00366 1.39e-06 2.05 285.20 9.27 275.93 1280 0.000872 3.31e-07 2.07 567.11 8.94 558.17 2560 0.000227 8.61e-08 1.94 1140.59 8.49 1132.10 $K$ $L^{\infty}$ error Relative error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) hIFE2 20 4.19 0.000397 - 11.11 0.00 0.28 40 1.05 0.000397 2.00 11.96 11.39 0.57 80 0.261 9.91e-05 2.00 11.61 10.70 0.91 160 0.0652 2.47e-05 2.00 12.36 10.37 1.99 320 0.0162 6.14e-06 2.01 13.90 9.84 4.06 640 0.00397 1.51e-06 2.03 17.65 9.43 8.22 1280 0.000971 3.68e-07 2.03 25.08 8.88 16.20 2560 0.000256 9.72e-08 1.92 40.83 8.45 32.38 $K$ $L^{\infty}$ error Relative error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) fEIF2 20 1.49e+29 4.43e+25 - 12.42 11.96 0.46 40 2.9e+48 8.65e+44 $-$64.08 12.61 11.78 0.83 80 6.04e+73 1.8e+70 $-$84.11 13.07 11.46 1.61 160 2.27e+96 6.77e+92 $-$74.99 14.43 11.20 3.23 320 1.92e+79 5.71e+75 56.72 17.20 10.59 6.61 640 0.251 7.48e-05 265.37 23.47 9.93 13.54 1280 0.119 3.54e-05 1.08 35.96 9.57 26.39 2560 0.0603 1.8e-05 0.98 62.05 9.08 52.97
 $K$ $L^{\infty}$ error Relative error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) IIF2 20 10 0.00381 - 5.32 5.25 0.07 40 4.81 0.00182 1.06 5.07 4.91 0.16 80 2.32 0.000881 1.05 5.09 4.78 0.31 160 1.12 0.000425 1.05 5.07 4.44 0.63 320 0.534 0.000203 1.07 5.24 3.90 1.34 640 0.251 9.51e-05 1.09 5.92 3.40 2.52 1280 0.115 4.34e-05 1.13 7.90 2.92 4.98 2560 0.0503 1.91e-05 1.19 12.84 2.55 10.29 $K$ $L^{\infty}$ error Relative error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) iETD2 20 3.99 0.00151 - 19.63 10.88 8.75 40 0.994 0.000377 2.00 28.60 10.80 17.80 80 0.248 9.41e-05 2.00 46.92 10.76 36.16 160 0.0617 2.34e-05 2.01 80.10 10.41 69.69 320 0.0152 5.76e-06 2.02 148.60 9.80 138.80 640 0.00366 1.39e-06 2.05 285.20 9.27 275.93 1280 0.000872 3.31e-07 2.07 567.11 8.94 558.17 2560 0.000227 8.61e-08 1.94 1140.59 8.49 1132.10 $K$ $L^{\infty}$ error Relative error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) hIFE2 20 4.19 0.000397 - 11.11 0.00 0.28 40 1.05 0.000397 2.00 11.96 11.39 0.57 80 0.261 9.91e-05 2.00 11.61 10.70 0.91 160 0.0652 2.47e-05 2.00 12.36 10.37 1.99 320 0.0162 6.14e-06 2.01 13.90 9.84 4.06 640 0.00397 1.51e-06 2.03 17.65 9.43 8.22 1280 0.000971 3.68e-07 2.03 25.08 8.88 16.20 2560 0.000256 9.72e-08 1.92 40.83 8.45 32.38 $K$ $L^{\infty}$ error Relative error Order CPU time (s) CPU time 1 (s) CPU time 2 (s) fEIF2 20 1.49e+29 4.43e+25 - 12.42 11.96 0.46 40 2.9e+48 8.65e+44 $-$64.08 12.61 11.78 0.83 80 6.04e+73 1.8e+70 $-$84.11 13.07 11.46 1.61 160 2.27e+96 6.77e+92 $-$74.99 14.43 11.20 3.23 320 1.92e+79 5.71e+75 56.72 17.20 10.59 6.61 640 0.251 7.48e-05 265.37 23.47 9.93 13.54 1280 0.119 3.54e-05 1.08 35.96 9.57 26.39 2560 0.0603 1.8e-05 0.98 62.05 9.08 52.97
Numerical a priori error in applying hIFE2 to a one-dimensional reaction–diffusion system with (A) $f(u,x,t) = \cos u+t$ for the decomposition (48) and (49) and (B) $f(u,x,t) = (t+1)\cos (xu)+xe^t$ for the decomposition (51) and (52). The a priori error is defined by $\|u^N-u^{N/2}\|_{\infty}$, where $N$ is the number of grid points in the spatial discretization. The simulations are run through time $T = 1$ with $\Delta x = \frac{\pi}{2N}$ and $\Delta t = 0.1\Delta x$
 (A) Decomposition (48) Decomposition (49) $N$ A priori error Order A priori error Order 16 0.00103 - 0.00102 - 32 0.000532 0.95 0.000255 2.00 64 0.000328 0.70 6.37e-05 2.00 128 8.21e-05 2.00 1.59e-05 2.00 256 0.000196 $-$1.25 3.98e-06 2.00 512 0.000106 0.88 9.95e-07 2.00 1024 8.69e-06 3.61 2.49e-07 2.00 2048 3.37e-05 $-$1.96 6.19e-08 2.01 4096 1.75e-05 0.95 1.42e-08 2.12 (B) Decomposition (51) Decomposition (52) N A priori error Order A priori error Order 16 0.00979 - 0.00977 - 32 0.00245 2.00 0.00245 1.99 64 6.67e-04 1.88 0.000614 2.00 128 1.67e-04 2.00 0.000153 2.00 256 3.94e-04 -1.24 3.83e-05 2.00 512 2.13e-04 0.89 9.59e-06 2.00 1024 1.76e-05 3.60 2.4e-06 2.00 2048 6.75e-05 -1.94 5.99e-07 2.00 4096 3.50e-05 0.95 1.48e-07 2.02
 (A) Decomposition (48) Decomposition (49) $N$ A priori error Order A priori error Order 16 0.00103 - 0.00102 - 32 0.000532 0.95 0.000255 2.00 64 0.000328 0.70 6.37e-05 2.00 128 8.21e-05 2.00 1.59e-05 2.00 256 0.000196 $-$1.25 3.98e-06 2.00 512 0.000106 0.88 9.95e-07 2.00 1024 8.69e-06 3.61 2.49e-07 2.00 2048 3.37e-05 $-$1.96 6.19e-08 2.01 4096 1.75e-05 0.95 1.42e-08 2.12 (B) Decomposition (51) Decomposition (52) N A priori error Order A priori error Order 16 0.00979 - 0.00977 - 32 0.00245 2.00 0.00245 1.99 64 6.67e-04 1.88 0.000614 2.00 128 1.67e-04 2.00 0.000153 2.00 256 3.94e-04 -1.24 3.83e-05 2.00 512 2.13e-04 0.89 9.59e-06 2.00 1024 1.76e-05 3.60 2.4e-06 2.00 2048 6.75e-05 -1.94 5.99e-07 2.00 4096 3.50e-05 0.95 1.48e-07 2.02
Numerical errors in the maximum norm for hIFE2 applied to the example in Section 5.4. The spatial resolution is $\Delta x = \frac{\pi}{2N}$ in all three dimensions, the time step is $\Delta t = 0.1\Delta x$, the ending time is $T = 1$, and the coefficients are $d_1 = d_2 = d_3 = 1$ and $r = -1$
 $N\times N\times N$ $L^{\infty}$ error Order $4\times 4 \times 4$ 1.33e-03 - $8\times 8 \times 8$ 3.28e-04 2.02 $16\times 16 \times 16$ 8.17e-05 2.01 $32\times 32\times 32$ 2.04e-05 2.00 $64\times 64 \times 64$ 5.10e-05 2.00 $128\times 128 \times 128$ 1.27e-06 2.00
 $N\times N\times N$ $L^{\infty}$ error Order $4\times 4 \times 4$ 1.33e-03 - $8\times 8 \times 8$ 3.28e-04 2.02 $16\times 16 \times 16$ 8.17e-05 2.01 $32\times 32\times 32$ 2.04e-05 2.00 $64\times 64 \times 64$ 5.10e-05 2.00 $128\times 128 \times 128$ 1.27e-06 2.00
A summary of the four methods: for their A-stability, and the restriction on $\Delta t$ to exhibit second order, with explicitly time-dependent reactions or nonhomogeneous boundary conditions
 Method A-stability $\Delta t$ to exhibit second-order accuracy Time-dependent reactions Nonhomogeneous BCs IIF2 Yes $\mathcal{O}(\Delta x^2)$ $\leq \mathcal{O}(\Delta x^2)$ iETD2 Yes $\mathcal{O}(1)$ - fEIF2 No $\mathcal{O}(1)$ $\mathcal{O}(1)$ hIFE2 Yes $\mathcal{O}(1)$ $\mathcal{O}(1)$ hIFE2 (transformed) Yes $\mathcal{O}(1)$ $\mathcal{O}(1)$
 Method A-stability $\Delta t$ to exhibit second-order accuracy Time-dependent reactions Nonhomogeneous BCs IIF2 Yes $\mathcal{O}(\Delta x^2)$ $\leq \mathcal{O}(\Delta x^2)$ iETD2 Yes $\mathcal{O}(1)$ - fEIF2 No $\mathcal{O}(1)$ $\mathcal{O}(1)$ hIFE2 Yes $\mathcal{O}(1)$ $\mathcal{O}(1)$ hIFE2 (transformed) Yes $\mathcal{O}(1)$ $\mathcal{O}(1)$
A comparison of the computational complexity between the IIF2, iETD2, hIFE2, and fEIF2 methods
 Operations per iteration Total complexity (ratio) IIF2 $\mathcal{O}(N^2)$ 1 iETD2 $\mathcal{O}(kN^3)$ $\mathcal{O}(kN)$ hIFE2 $\mathcal{O}(3N^2)$ 3 fEIF2 $\mathcal{O}(5N^2)$ 5
 Operations per iteration Total complexity (ratio) IIF2 $\mathcal{O}(N^2)$ 1 iETD2 $\mathcal{O}(kN^3)$ $\mathcal{O}(kN)$ hIFE2 $\mathcal{O}(3N^2)$ 3 fEIF2 $\mathcal{O}(5N^2)$ 5
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