Exact difference schemes for a two-dimensional convection-diffusion-reaction equation

The initial-boundary value problem for a nonlinear two-dimensional convection-diffusion-reaction equation (1)∂u∂t+∑α=12aα∂u∂xα=∑α=12∂∂xα(kα(u)∂u∂xα)+λu,(x,t)∈Ω×(0,T],(2)u(x,0)=u0(x),x∈Ω¯,u|x∈∂Ω=g(x,t),(x,t)∈∂Ω×(0,T], is considered. The traveling-wave solutions of the problem are under special consideration. The problem is approximated by the difference scheme, which is exact for this type of solutions. Presented numerical experiments illustrate the theoretical results investigated in the paper.