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This paper studies the numerical solution of traveling singular sourcesproblems. In such problems, a big challenge is the sources move with differentspeeds, which are described by some ordinary differential equations. Apredictor-corrector algorithm is presented to simulate the position of singularsources. Then a moving mesh method in conjunction with domain decomposition isderived for the underlying PDE. According to the positions of the sources, thewhole domain is splitted into several subdomains, where moving mesh equationsare solved respectively. On the resulting mesh, the computation of jump$[\dot{u}]$ is avoided and the discretization of the underlying PDE is reducedinto only two cases. In addition, the new method has a desired second-order ofthe spatial convergence. Numerical examples are presented to illustrate theconvergence rates and the efficiency of the method. Blow-up phenomenon is alsoinvestigated for various motions of the sources.

Numerical investigations of traveling singular sources problems via moving mesh method