Numerical simulation of blowup in nonlocal reaction-diffusionequations using a moving mesh method

摘要

In this paper we implement the moving mesh PDE method for simulating the blowup inreaction-diffusion equations with temporal and spacial nonlinear nonlocal terms. By atime-dependent transformation, the physical equation is written into a Lagrangian formwith respect to the computational variables. The time-dependent transformation functionsatisfies a parabolic partial differential equation — usually called moving mesh PDE(MMPDE). The transformed physical equation and MMPDE are solved alternately by centralfinite difference method combined with a backward time-stepping scheme. The integrationtime steps are chosen to be adaptive to the blowup solution by employing a simple andefficient approach. The monitor function in MMPDEs plays a key role in the performanceof the moving mesh PDE method. The dominance of equidistribution is utilized to selectthe monitor functions and a formal analysis is performed to check the principle. A varietyof numerical examples show that the blowup profiles can be expressed correctly in thecomputational coordinates and the blowup rates are determined by the tests.