Article Contents
Article Contents

# Numerical treatment of Gray-Scott model with operator splitting method

• This article focuses on the numerical solution of a classical, irreversible Gray Scott reaction-diffusion system describing the kinetics of a simple autocatalytic reaction in an unstirred ow reactor. A novel finite element numerical scheme based on B-spline collocation method is developed to solve this model. Before applying finite element method, "strang splitting" idea especially popularized for reaction-diffusion PDEs has been applied to the model. Then, using the underlying idea behind finite element approximation, the domain of integration is partitioned into subintervals which is sought as the basis for the B-spline approximate solution. Thus, the partial derivatives are transformed into a system of algebraic equations. Applicability and accuracy of this method is justified via comparison with the exact solution and calculating both the error norms $L_2$ and $L_\infty$. Numerical results arising from the simulation experiments are also presented.

Mathematics Subject Classification: Primary: 65L60, 35Q68; Secondary: 65D07.

 Citation:

• Figure 1.  Numerical simulation of Gray Scott model

Figure 2.  Numerical simulation of Gray Scott model

Figure 3.  Numerical simulation of Gray Scott model for $u_{apprx}\left( x, t\right)$ and $v_{apprx}\left( x, t\right)$

Table 1.  Gray Scott model: The error norms for $\Delta t = 0.01$ and various values of $h$ at $T = 1$

 $u\left( x, t\right)$ $v\left( x, t\right)$ $h$ $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ 0.1 16.255794672 8.8693520879 16.255794674 8.8693521167 0.05 4.7212245929 3.5788660872 4.7212245653 3.5788660765 0.025 3.1114886810 3.6707532707 3.1114889322 3.6707531289 0.0125 3.3850799985 9.7614962725 3.3850788862 9.7614970661

Table 2.  Gray Scott model: The error norms for $\Delta t = 0.0025$ and various values of $h$ at $T = 1$

 $u\left( x, t\right)$ $v\left( x, t\right)$ $h$ $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ 0.1 16.339991746 8.7969669421 16.339991702 8.7969670275 0.05 4.8404628306 3.5790496331 4.8404627644 3.5790495422 0.025 2.9993515295 3.6711268925 2.9993514761 3.6711268910 0.0125 2.9168925317 3.7175342072 2.9168926962 3.7175350072

Table 3.  Gray Scott model: The error norms for $\Delta t = 0.00125$ and various values of $h$ at $T = 1$

 $u\left( x, t\right)$ $v\left( x, t\right)$ h $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ $L_{2}\times10^{6}$ $L_{\infty }\times 10^{6}$ 0.1 16.374721220 8.7933451145 16.374721299 8.7933453672 0.05 4.9049725941 3.5790587130 4.9049726730 3.5790587406 0.025 3.0513500430 3.6711456886 3.0513501767 3.6711458248 0.0125 3.0513500430 3.6711456886 3.0513501767 3.6711458248

Table 4.  Gray Scott model: The error norms for $\Delta t = 0.001$ and various values of $h$ at $T = 1$

 $u\left( x, t\right)$ $v\left( x, t\right)$ h $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ 0.1 16.382270949 8.7929105788 16.382270862 8.7929107112 0.05 4.9186006922 3.5790599957 4.9186005560 3.5790598445 0.025 3.0624459954 3.6711482907 3.0624458638 3.6711480981 0.0125 2.9466605008 3.7175784792 2.9466601800 3.7175779681
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