Numerical simulation of blowup in nonlocal reaction-diffusion equations using a moving mesh method. (English) Zbl 1166.65067
Summary: We implement the moving mesh partial differential equation (PDE) method for simulating the blowup in reaction-diffusion equations with temporal and spacial nonlinear nonlocal terms. By a time-dependent transformation, the physical equation is written into a Lagrangian form with respect to the computational variables. The time-dependent transformation function satisfies a parabolic partial differential equation – usually called moving mesh PDE (MMPDE).
The transformed physical equation and MMPDE are solved alternately by a central finite difference method combined with a backward time-stepping scheme. The integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach. The monitor function in MMPDEs plays a key role in the performance of the moving mesh PDE method. The dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle. A variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests.
The transformed physical equation and MMPDE are solved alternately by a central finite difference method combined with a backward time-stepping scheme. The integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach. The monitor function in MMPDEs plays a key role in the performance of the moving mesh PDE method. The dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle. A variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45K05 | Integro-partial differential equations |
| 45G10 | Other nonlinear integral equations |
Keywords:
blowup; nonlocal reaction-diffusion equations; integro-differential equations; moving mesh methods; finite difference method; numerical examplesReferences:
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