Numerical methods for coupled problems involving reaction-diffusion equations

本研究的目的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，这些系统需要在不同位置使用不同的模型。从一种模型到另一种模型的切换通常是通过尖锐的界面发生的，无论是固定的还是移动的，要么以规定的速度进行，要么取决于溶液的速度。这种现象的相关例子包括：(1)心脏动作电位在被细胞外空间包围的肌细胞网络上传播；(2)电势在嵌入躯干的固定或移动的心脏上传播；(3)物种在移动的有利斑块上的生态扩散，有利于在不利环境中生长，从而导致灭绝。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组，通过边界条件耦合的方程和/或界面上的微分方程组。在单个区域上数值求解反应扩散方程组已经是一个挑战，这是因为具有尖锐梯度的行波与静止解相邻的区域。求解这些方程的数值方法需要大量的计算资源，尤其是对于3D现象。例如，心脏中的电波需要在每个时间步长求解10^7到10^8个耦合方程，每个心跳超过10,000个时间步长。求解在不同子域上表示的、通过界面传输条件耦合的不同反应扩散方程，增加了以一种易于得到有效和准确的数值解的方式来描述相关问题的复杂性。如果界面在移动，这就更难了。我的研究计划将推进反应-扩散耦合方程问题的数值求解。更具体地说，我们将致力于这些问题的三个方面：(I)时间步长方法及其在空间网格自适应循环中的应用，以达到时间上的精度；(Ii)界面问题的公式和数值解，以便能够解决复杂的多物理问题；(Iii)误差估计和网格自适应，以达到空间上的精度。除了培训3名硕士和6名博士生外，该研究计划还将大大促进流行的反应扩散系统数值方法的进步。从更实际的角度来看，这将导致开发出高效和准确的心脏和空间人口动力学数值模型。数值模型是医学、工程和科学领域的重要决策工具，在很大程度上依赖于数值算法的不断完善。

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.