AI 中文总结
该研究提出一种基于常微分方程组的反应扩散系统建模新方法，其采用统计简化和几何概率原理描述扩散，可简化定性分析但效率提升仅适用于特定情况。
AI 中文摘要
我们提出一种基于常微分方程组的反应扩散系统建模新方法。与直线法等专用数值方法不同，该新方法在偏微分方程的模型层面作为纯粹替代方案。其描述与有限体积法高度相似，但不同之处在于，它采用统计简化和几何概率原理来描述扩散。该方法的主要目标是简化反应扩散系统的定性分析并提高数值模型实现的效率。第一个目标已成功达成，因为可利用经典动力学系统理论的工具对基于常微分方程组的模型动力学进行定性分析。第二个目标仅部分达成，因为在保持可接受精度的前提下，仅当分子初始分布为特定简单形式且扩散系数为特定值时，数值实现的效率提升才显著。此外，为制定实际应用的标准，我们单独评估了该新方法的建模误差。
英文摘要
We consider a new methodology for modelling the reaction-diffusion systems based on systems of ordinary differential equations. In contrary to the specialised numerical methods like straight line method, this new methodology is positioned as a pure alternative at the model level for partial differential equations. In its description, the new method is largely similar to the finite volume method, but unlike the latter, it uses statistical simplifications and principles of geometric probability to describe diffusion. The main objectives of this approach are to simplify the qualitative analysis of reaction-diffusion systems and to improve the efficiency of numerical model implementation. The first objective is successfully addressed, as it becomes possible to use the apparatus of classical dynamical systems theory for a qualitative analysis of model dynamics based on the systems of ordinary differential equations. The second objective is only partially addressed, as the gain in efficiency while maintaining acceptable accuracy for numerical implementation will be significant only for certain simple initial distribution of molecules and for specific diffusion coefficients. Furthermore, to formulate criteria for practical applicability, we separately evaluate the modelling error using this new methodology.
Journal refM. N. Nazarov, A new approach to the reaction-diffusion systems modelling, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, no. 4, 84-94
