Moving collocation method for a reaction-diffusion equation with a traveling heat source

摘要

We consider a reaction-diffusion equation with a traveling heat source on an unbounded domain. The numerical simulation of the problem is difficult because of the moving singularity, the blow-up phenomenon, and the delta function in the equation. Because we are only interested in the solution behavior near the heat source, we choose a bounded moving domain which contains the heat source and has the same speed as the source. Local absorbing boundary conditions are constructed on the boundaries of the moving domain. Then, we transform the moving domain to a fixed one. At last, a special moving collocation method is adopted. The new method is much simpler than the existing moving finite difference methods. Moreover, numerical experiments illustrate the accuracy and efficiency of our moving collocation method.