A fourth-order exponential time differencing scheme with real and distinct poles rational approximation for solving nonlinear reaction-diffusion systems
Student: Wisdom Kwame Attipoe
Advisor: Dr. Emmanuel O. Asante-Asamani
Ph.D Committee: Dr Kathleen Kavanagh, Dr. Guangming Yao, Dr. Suguang Xiao, Dr. Prashant Athavale
Thursday, May 1st, 2025
10:00 AM
CAMP 194
ABSTRACT
Reaction-diffusion systems are mathematical models that describe the spatio-temporal dynamics of chemical substances as they react and diffuse in a medium. They occur as models of biological pattern formation, pollutant transport in surface and groundwater, tumor angiogenesis and option pricing. Higher-order time discretization methods that can be coupled with efficient spatial discretization schemes are highly desirable for solving such problems. In this work, we develop a fourth-order exponential time differencing (ETD) Runge-Kutta scheme with a real and distinct poles (RDP) rational approximation to solve multidimensional nonlinear systems of reaction-diffusion equations (RDEs). The matrix exponentials in the scheme are approximated with RDP rational functions, which are L-acceptable, allowing for the smoothing of spurious oscillations in problems with mismatched initial and boundary conditions. Various nonlinear reaction-diffusion systems with Dirichlet and Neumann boundary conditions are used to empirically validate the order of convergence of the scheme and compare its performance with existing fourth-order schemes. Additionally, we show that a parallel implementation of the scheme can result in 2-4 times speed up in CPU time. In the future, we propose to rigorously prove the convergence of the scheme and develop a dimensional splitting version to further improve computational efficiency for multidimensional problems.