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准确有效的数值方法。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，这些方程被应用于具有空间异质性的系统，这些系统在不同的位置需要不同的模型。从一种模型到另一种模型的转换通常是通过固定或移动的尖锐界面进行的，要么具有规定的速度，要么具有依赖于解决方案的速度。这种现象的相关例子包括：(1)在细胞外空间包围的肌细胞网络上传播心脏动作电位；(2)电势在嵌入躯干的固定或移动心脏上的传播；(3)物种在不利环境中迁移到有利于生长的斑块上导致灭绝的生态扩散。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，通过边界条件耦合的方程和/或界面上的微分方程系统。求解单域上的反应扩散方程的数值系统已经是一个挑战，因为行波具有尖锐的梯度，邻近区域具有静态解。用数值方法求解这些方程需要大量的计算资源，特别是对于三维现象。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，每心跳超过10,000个时间步。求解在不同子域上表示并通过界面传输条件耦合的不同反应扩散方程，增加了用一种易于适用于高效和精确数值解的方法来表述相关问题的复杂性。如果界面是移动的，这就更加困难了。我的研究计划将推进具有耦合反应-扩散方程问题的数值解。更具体地说，我们将从三个方面研究这些问题：(1)时间步进方法及其在空间网格自适应环中的包含，以达到及时的精度；（二）界面问题的表述和数值解，能够解决复杂的多物理场问题；（三）误差估计和网格自适应以达到空间精度。除了培养3名硕士生和6名博士生外，该研究项目将为流行的反应扩散系统的数值方法的发展做出重大贡献。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、工程和科学领域重要的决策工具，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.