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. Author manuscript; available in PMC: 2020 May 13.
Published in final edited form as: Discrete Continuous Dyn Syst Ser B. 2019 Dec;24(12):6387–6417. doi: 10.3934/dcdsb.2019144

Table 5.

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 (Δx = π/2N), and run the simulation for K time steps (Δt = T/K). The error e is measured in the maximum norm, and the relative error is defined by e/max{∥UK∥∞, ∥VK∥∞}, where UK and VK 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∞ 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∞ 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∞ 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∞ 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