Numerical methods for coupled problems involving reaction-diffusion equations

我的研究目标是为涉及反应扩散方程的复杂时空现象提供精确有效的数值方法。这些方程通常用于描述空间分布的生物、化学和物理系统的时间演化。最近，这些方程已被应用到系统的空间异质性，需要在不同的位置不同的模型。从一个模型到另一个模型的转换通常是通过尖锐的界面发生的，固定的或移动的，要么是规定的速度，要么是依赖于解决方案的速度。这种现象的相关例子包括：（1）心脏动作电位在细胞外空间包围的肌细胞网络上的传播;（2）电位在嵌入躯干的固定或移动心脏上的传播;（3）物种在不利环境中移动有利于生长的良好斑块上的生态扩散导致灭绝。在这三个例子中，在尖锐界面两侧的子域上需要不同的偏微分方程，通过界面上的边界条件和/或微分方程系统耦合的方程。在单个区域上数值求解反应扩散方程组已经是一个挑战，因为行波具有尖锐的梯度，邻近区域具有静态解。求解这些方程的数值方法需要大量的计算资源，特别是对于3D现象。例如，心脏中的电波在每个时间步需要求解10^7到10^8个耦合方程，每个心跳超过10，000个时间步。求解在不同子域上表达的不同反应扩散方程，并通过界面传输条件耦合，这增加了以易于进行有效和准确的数值解的方式制定相关问题的复杂性。如果接口在移动，这就更加困难了。我的研究计划将推进耦合反应扩散方程问题的数值解。更具体地说，我们将致力于这些问题的三个方面：（I）时间步进方法及其在空间网格自适应循环中的包含，以达到时间上的准确性;（II）接口问题的公式化和数值解，以解决复杂的多物理问题;（III）误差估计和网格自适应，以达到空间上的准确性。除了培养3名硕士和6名博士生外，该研究计划还将为普遍反应扩散系统的数值方法的进步做出重大贡献。从更实际的角度来看，它将导致心脏和空间人口动态的有效和准确的数值模型的发展。数值模型是医学、工程和科学等领域的重要决策工具，而这些领域的研究在很大程度上依赖于数值算法的不断改进。

The goal of my research is to provide accurate and efficient numerical methods for complex spatio-temporal phenomena involving reaction-diffusion equations. These equations are commonly used to describe the time-evolution of spatially distributed biological, chemical and physical systems. More recently these equations have been applied to systems with spatial heterogeneities that require different models in different locations. The switch from one model to the other is often occurring through sharp interfaces, fixed or moving, either with a prescribed or solution-dependent velocity. Relevant examples of such phenomena include: (1) the propagation of the cardiac action potential over a network of myocytes surrounded by extra-cellular space; (2) the propagation of the electrical potential over either a fixed or moving heart embedded in a torso; (3) the ecological dispersal of a species over moving good patches favorable to growth in an unfavorable environment leading to extinction. In these three examples, different partial differential equations are needed on the subdomains on both sides of sharp interfaces, equations which are coupled through boundary conditions and/or systems of differential equations on the interfaces. Solving numerically systems of reaction-diffusion equations on a single domain is already a challenge due to traveling waves with sharp gradients neighboring zones with quiescent solutions. Numerical methods to solve these equations need prohibitive computational resources, especially for 3D phenomena. For example, electrical waves in the heart require 10^7 to 10^8 coupled equations to be solved at each time step, over 10,000 time steps per heart beat. Solving different reaction-diffusion equations expressed on different subdomains and coupled through interface transmission conditions raises the complication of formulating the related problem in a way easily amenable to efficient and accurate numerical solutions. This is even more difficult if the interface is moving. My research program will advance the numerical solution of problems with coupled reaction-diffusion equations. More specifically, we will work on three aspects of these problems: (I) Time-stepping methods and their inclusion in spatial mesh adaptation loops to reach accuracy in time; (II) The formulation and numerical solution of interface problems to be able to solve complex multi-physics problems; (III) Error estimation and mesh adaptation to reach accuracy in space. Aside from training 3 MSc and 6 PhD students, the research program will significantly contribute to the advancement of numerical methods for prevalent reaction-diffusion systems. From a more practical standpoint, it will lead to the development of efficient and accurate numerical models of the heart and spatial population dynamics. Numerical models are important decision tools in medicine, engineering and science, which largely rely on the on-going improvement of numerical algorithms.