An Efficient Explicit Finite-Difference Scheme for Simulating Coupled Biomass Growth on Nutritive Substrates
An Efficient Explicit Finite-Difference Scheme for Simulating Coupled Biomass Growth on Nutritive Substrates
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DOI：
10.1155/2015/708497
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发表时间：
2015-01
影响因子：
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通讯作者：
G. Sun;G. Liu;andM. Li
中科院分区：
文献类型：
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作者：
G. Sun;G. Liu;andM. Li
提出了一种新颖的显式有限差分（FD）方法来模拟微生物菌落在营养物基质下的正向有界发育过程，该过程由非线性抛物型偏微分方程（PDE）系统控制。我们的显式FD方案的独特设计方式是将原始偏微分方程中的非线性项转换为代数方程组中的离散线性项组，可以非常有效地求解，同时保证解的稳定性和有界性。这是通过以下方式实现的：(1) 对时间和空间变化中的扩散函数项进行适当的交织 FD 近似设计，以及 (2) 通过建立理论稳定性标准来控制时间步长。详细的理论稳定性分析表明我们的 FD 方法确实是稳定的。我们的例子验证了数值解可以保证非负且有界以模拟实际物理现象。还提供了数值示例来证明所提出方案的效率。该方案适用于在生物膜动力学研究中求解类似的偏微分方程组。
A novel explicit finite-difference (FD) method is presented to simulate the positive and bounded development process of a microbial colony subjected to a substrate of nutrients, which is governed by a nonlinear parabolic partial differential equations (PDE) system. Our explicit FD scheme is uniquely designed in such a way that it transfers the nonlinear terms in the original PDE into discrete sets of linear ones in the algebraic equation system that can be solved very efficiently, while ensuring the stability and the boundedness of the solution. This is achieved through (1) a proper design of intertwined FD approximations for the diffusion function term in both time and spatial variations and (2) the control of the time-step through establishing theoretical stability criteria. A detailed theoretical stability analysis is conducted to reveal that our FD method is indeed stable. Our examples verified the fact that the numerical solution can be ensured nonnegative and bounded to simulate the actual physics. Numerical examples have also been presented to demonstrate the efficiency of the proposed scheme. The present scheme is applicable for solving similar systems of PDEs in the investigation of the dynamics of biological films.