Adaptive finite volume approximations for weakly coupled convection dominated parabolic systems. (English) Zbl 1007.65059
Consider the following convection-diffusion Cauchy problem
\[
c_t+ \nabla\cdot({\mathbf u}^i(x,t) f^i(c^i))- \nabla\cdot(D^i(x, t)\nabla c^i)+ \lambda^i(x,t,{\mathbf c})= 0,\;c^i(x, 0)= c^i_0,
\]
\(i= 1,2,\dots, M\). Here \({\mathbf u}^i\) is a vector field and \({\mathbf f}\) a flux function. The system is weakly coupled because the unknown functions \(c^i\) are only coupled through \(\lambda^i\). The authors are studying the case when convecting fluxes dominate the diffusive terms and define weak solutions for this system which they assume to exist. They present finite volume schemes on unstructured grids to approximate the solution and give an \(L^1\)-error a posteriori estimate. Numerical experiments on environmental and combustion problems are also presented.
Reviewer: Erwin Schechter (Kaiserslautern)
MSC:
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
35K57 | Reaction-diffusion equations |