A Multiscale Algorithm for Solving Reaction-Diffusion Equations
by
Jianping Zhu
Mississippi State University
Coauthors: Lilun Cao, Pasquale Cinnella

Many engineering and physical applications, such as combustion and reactive flow, are described by coupled systems of partial differential equations involving significantly different time scales. For example, in the study of reaction and diffusion processes involving multiple chemical components, the diffusion and reaction rates for different components can be significantly different. A typical equation system describing such a process is the reaction-diffusion equation

ut = D uxx + f (x, t, u)     (1)

where u is a vector of p state variables, D is a diagonal matrix with the diffusion coefficients of different components of u on the diagonal, and f (x, u) is a nonlinear reaction term that depends on the space and state variables. If we use an implicit time integration algorithm, such as the Crank-Nicolson scheme, to compute the numerical solution of (1) at different time steps, we will have the following nonlinear algebraic equations:

AUn+1=BUn + Fn+1

where A and B are matrices, Uⁿ is the vector containing discrete values of state variables u at all grid points at time t_n, and Fⁿ⁺¹ is the vector that in general depends on Uⁿ⁺¹ nonlinearly. A linearization method, such as the Newton's method, is usually needed to solve this system of nonlinear algebraic equations. Furthermore, since all equations in (1) are coupled, the numerical solutions to such a system are usually calculated using a time step determined by the most restrictive temporal scale in the system for accuracy considerations. We demonstrate in this paper that this time step could be excessively small and unnecessary in many situations, and discuss a more efficient time integration method that does not require linearization procedure and uses different time steps for different equations depending on their respective time scales. The basic idea is to decouple the system of equations and then apply different time steps to different equations depending on their time scales. This will also eliminate the need for linearization. The challenge is to maintain stability and accuracy of the original implicit algorithm. Detailed stability and truncation error analyses will be presented in the paper to prove that, by using proper interpolations between time steps, one can maintain the stability and accuracy of the original implicit algorithm while using different time steps for different equations in a coupled system.

Numerical results from test problems will be discussed to demonstrate significant improvement in computational efficiency.

Date received: February 9, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caav-09.