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Article Contents

# Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries

• * Corresponding author: Xinfeng Liu
• The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population. There are several numerical difficulties to efficiently handle such systems. Firstly extremely small time steps are usually demanded due to the stiffness of the system. Secondly it is always difficult to efficiently and accurately handle the moving boundaries. To overcome these difficulties, we first transform the one-dimensional problem with a moving boundary into a system with a fixed computational domain, and then introduce four different temporal schemes: Runge-Kutta, Crank-Nicolson, implicit integration factor (IIF) and Krylov IIF for handling such stiff systems. Numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches, and it can be shown that Krylov IIF is superior to other three approaches in terms of stability and efficiency by direct comparison.

Mathematics Subject Classification: Primary: 65N06, 65N40; Secondary: 92D25.

 Citation:

• Figure 1.  Error of U as a function of time step sizes

Figure 2.  Error of H as a function of time step sizes

Figure 3.  Solution $U$ and $H$ for the large diffusion system

Figure 4.  Solution $U$ and $H$ for the stiff system

Table 1.  Convergence test of Runge-Kutta method

 $N_z\times N_t$ $L_{\infty} Error$ Order $L_2 Error$ Order Accuracy test of W 26$\times$5e4 1.85e-04 1.32e-04 51$\times$1e5 4.62e-05 2.00 3.28e-05 2.01 101$\times$2e5 1.16e-05 1.99 8.22e-06 2.00 201$\times$4e5 2.89e-06 2.01 2.04e-06 2.01 401$\times$8e5 6.39e-07 2.18 4.50e-07 2.18 801$\times$16e5 Reference Accuracy test of G 26$\times$5e4 2.66e-04 6.09e-06 51$\times$1e5 6.65e-05 2.00 1.52e-06 2.00 101$\times$2e5 1.68e-05 1.99 3.85e-07 1.98 201$\times$4e5 4.20e-06 2.00 9.65e-08 2.00 401$\times$8e5 9.38e-07 2.16 2.17e-08 2.15 801$\times$16e5 Reference

Table 2.  Convergence test of Crank-Nicolson method

 $N_z\times N_t$ $L_{\infty} Error$ Order $L_2 Error$ Order Accuracy test of W 26$\times$5e4 1.85e-04 2.68e-04 51$\times$1e5 4.65e-05 2.00 3.30e-05 1.99 101$\times$2e5 1.18e-05 1.98 8.30e-06 1.99 201$\times$4e5 2.92e-06 2.01 2.06e-06 2.01 401$\times$8e5 6.30e-07 2.21 4.38e-07 2.23 801$\times$16e5 Reference Accuracy test of G 26$\times$5e4 1.33e-04 6.13e-05 51$\times$1e5 6.72e-05 2.01 1.54e-05 1.99 101$\times$2e5 1.71e-05 1.98 3.92e-06 1.97 201$\times$4e5 4.30e-06 1.99 9.90e-07 1.99 401$\times$8e5 9.30e-07 2.21 2.15e-07 2.20 801$\times$16e5 Reference

Table 3.  Convergence test of IIF2 method

 $N_z\times N_t$ $L_{\infty} Error$ Order $L_2 Error$ Order Accuracy test of W 26$\times$5e4 1.82e-04 1.31e-04 51$\times$1e5 4.51e-05 2.02 3.20e-05 2.03 101$\times$2e5 1.11e-05 2.02 7.84e-06 2.03 201$\times$4e5 2.65e-06 2.07 1.86e-06 2.07 401$\times$8e5 5.34e-07 2.31 3.73e-07 2.32 801$\times$16e5 Reference Accuracy test of G 26$\times$5e4 2.65e-04 6.07e-06 51$\times$1e5 6.58e-05 2.01 1.51e-06 2.01 101$\times$2e5 1.64e-05 2.00 3.78e-07 2.00 201$\times$4e5 4.00e-06 2.04 9.26e-08 2.03 401$\times$8e5 8.35e-07 2.26 1.94e-08 2.26 801$\times$16e5 Reference

Table 4.  Convergence test of Krylov IIF2 method

 $N_z\times N_t$ $L_{\infty} Error$ Order $L_2 Error$ Order Accuracy test of W 26$\times$5e4 1.82e-04 1.31e-04 51$\times$1e5 4.51e-05 2.02 3.20e-05 2.03 101$\times$2e5 1.11e-05 2.02 7.84e-06 2.03 201$\times$4e5 2.65e-06 2.07 1.86e-06 2.07 401$\times$8e5 5.30e-07 2.32 3.72e-07 2.32 801$\times$16e5 Reference Accuracy test of G 26$\times$5e4 2.65e-04 6.07e-06 51$\times$1e5 6.58e-05 2.01 1.51e-06 2.01 101$\times$2e5 1.64e-05 2.00 3.78e-07 2.00 201$\times$4e5 4.00e-06 2.04 9.26e-08 2.03 401$\times$8e5 8.40e-07 2.25 1.94e-08 2.26 801$\times$16e5 Reference

Table 5.  Errors and order of accuracy in time for three stable schemes: Crank-Nicolson, IIF2 and Krylov IIF2

 $\triangle t$ Crank-Nicolson IIF2 Krylov IIF2 Accuracy test of W $L_{\infty}$ error Order $L_{\infty}$ error Order $L_{\infty}$ error Order $8.0\times10^{-5}$ 1.54e-8 - 6.50e-8 - 6.50e-8 - $4.0\times10^{-5}$ 4.09e-9 1.91 2.23e-8 1.55 2.23e-8 1.55 $2.0\times10^{-5}$ 1.05e-9 1.97 6.28e-9 1.83 6.28e-9 1.83 $1.0\times10^{-5}$ 3.22e-10 1.70 1.30e-9 2.27 1.30e-9 2.27 $5.0\times10^{-6}$ 8.14e-11 1.98 2.85e-10 2.20 2.85e-10 2.20 $2.5\times10^{-6}$ Reference Reference Reference Accuracy test of G $L_{\infty}$ error Order $L_{\infty}$ error Order $L_{\infty}$ error Order $8.0\times10^{-5}$ 4.16e-7 - 8.49e-7 - 8.49e-7 - $4.0\times10^{-5}$ 1.44e-7 1.53 2.81e-7 1.59 2.81e-7 1.59 $2.0\times10^{-5}$ 4.86e-8 1.57 9.22e-8 1.61 9.22e-8 1.61 $1.0\times10^{-5}$ 1.54e-8 1.66 2.88e-8 1.68 2.88e-8 1.68 $5.0\times10^{-6}$ 3.92e-9 1.97 7.27e-9 1.99 7.27e-9 1.99 $2.5\times10^{-6}$ Reference Reference Reference

Table 6.  Efficiency test for the large diffusion system

 $\triangle t=10^{-4}$ $N_z=1001$ $N_z=2001$ $N_z=4001$ Crank-Nicolson 16.576 75.092 395.512 Krylov IIF2 14.595 59.566 295.958 IIF2 211.227 1099.695 9694.277

Table 7.  Efficiency test for the stiff system

 $\triangle t=10^{-4}$ $N_z=1001$ $N_z=2001$ $N_z=4001$ Crank-Nicolson 30.149 141.601 760.832 Krylov IIF2 16.706 71.501 346.278 IIF2 389.514 1877.676 22626.109

Figures(4)

Tables(7)