Devising efficient numerical methods for oscillating patterns in reaction-diffusion systems

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• 作者：Giuseppina Settanni^a ; ^{giuseppina.settanni@uniba.it" class="auth_mail" title="E-mail the corresponding author} ; Ivonne Sgura^b ; ^{ivonne.sgura@unisalento.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Turing&ndash ; Hopf patterns ; Schnakenberg model ; High order finite differences ; IMEX methods ; ADI methods ; Symplectic schemes
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：292
• 期：Complete
• 页码：674-693
• 全文大小：3316 K

文摘

In this paper, we consider the numerical approximation of a reaction–diffusion system 2D in space whose solutions are patterns oscillating in time or both in time and space. We present a stability analysis for a linear test heat equation in terms of the diffusion k to view the MathML source">d$d$ and of the reaction timescales given by the real and imaginary parts k to view the MathML source">伪$伪$ and k to view the MathML source">尾$尾$ of the eigenvalues of k to view the MathML source">J(P_e)$J\left(P_e\right)$, the Jacobian of the reaction part at the equilibrium point k to view the MathML source">P_e$P_e$. Focusing on the case 952317a7b6196add05c95b433db2e1" title="Click to view the MathML source">伪=0,尾≠0$伪=0,尾\ne 0$, we obtain stability regions in the plane k to view the MathML source">(尉,谓)$(尉,谓)$, where i96" class="mathmlsrc">i96.gif&_user=111111111&_pii=S0377042715002708&_rdoc=1&_issn=03770427&md5=46f50c991b52392b6944393354a100f9" title="Click to view the MathML source">尉=位(h;d)h_t, k to view the MathML source">谓=尾h_t$谓=尾h_t$, k to view the MathML source">h_t$h_t$ time stepsize, k to view the MathML source">位$位$ lumped diffusion scale depending also from the space stepsize k to view the MathML source">h$h$ and from the spectral properties of the discrete Laplace operator arising from the semi-discretization in space. In space we apply the Extended Central Difference Formulas (ECDFs) of order k to view the MathML source">p=2,4,6$p=2,4,6$. In time we approximate the diffusion part in implicit way and the reaction part by a selection of integrators: the Explicit Euler and ADI methods, the symplectic Euler and a partitioned Runge–Kutta method that are symplectic in the absence of diffusion. Hence, by estimating k to view the MathML source">位$位$, for each method we derive stepsize restrictions k to view the MathML source">h_t猹匜_met(h;d,尾,p) in terms of the stability curve 951c06751ee72c365726bb4589aa7" title="Click to view the MathML source">F_met$F_{met}$ depending on diffusion and reaction timescales and from the approximation order in space. For the same schemes, we provide also a dispersion error analysis. We present numerical simulations for the test heat equation and for the Lotka–Volterra PDE system with solutions oscillating only in time for the presence of a centre-type dynamics. In these cases, the implicit-symplectic schemes provide the best choice. We solve also the Schnakenberg model with spatial patterns oscillating in space and time in the presence of an attractive limit cycle due to the Turing–Hopf instability. In this case, all schemes attain closed orbits in the phase space, but the Explicit ADI method is the best choice from the computational point of view.