Matrix-oriented discretization methods for reaction-diffusion PDEs: Comparisons and applications
Matrix-oriented discretization methods for reaction-diffusion PDEs: Comparisons and applications
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反应扩散偏微分方程的面向矩阵的离散方法：比较和应用
DOI：
10.1016/j.camwa.2019.10.020
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
V. Simoncini
中科院分区：
文献类型：
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作者：
M. D’Autilia;I. Sgura;V. Simoncini
反应扩散偏微分方程组广泛应用于生命科学和物理化学现象的建模。特别是，扩散和非线性动力学之间的耦合可以导致所谓的图灵不稳定性，从而产生各种空间图案（如条纹，斑点，条纹等）。获得大时间间隔的稳态解。为了捕捉模式本身的形态特性，可能需要非常精细的空间离散化，由于计算成本过高，限制了标准（基于矢量的）ODE求解器的使用。通过利用扩散矩阵的结构，我们表明，基于矩阵的版本的时间积分，如隐式显式（IMEX）和指数计划，允许更精细的问题离散化。我们通过数值求解Schnakenberg模型来说明我们的研究结果，该模型是具有图灵模式解决方案的RD-PDE系统的原型，以及描述电池充电过程中金属生长的DIB形态化学模型。
Systems of reaction–diffusion partial differential equations (RD-PDEs) are widely applied for modeling life science and physico-chemical phenomena. In particular, the coupling between diffusion and nonlinear kinetics can lead to the so-called Turing instability, giving rise to a variety of spatial patterns (like labyrinths, spots, stripes, etc.) attained as steady state solutions for large time intervals. To capture the morphological peculiarities of the pattern itself, a very fine space discretization may be required, limiting the use of standard (vector-based) ODE solvers in time because of excessive computational costs. By exploiting the structure of the diffusion matrix, we show that matrix-based versions of time integrators, such as Implicit–Explicit (IMEX) and exponential schemes, allow for much finer problem discretizations. We illustrate our findings by numerically solving the Schnakenberg model, prototype of RD-PDE systems with Turing pattern solutions, and the DIB-morphochemical model describing metal growth during battery charging processes.