Numerical methods for coupled problems involving reaction-diffusion equations

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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.