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Eulerian-Lagrangian method of fundamental solutions for multi-dimensional advection-diffusion equation. (English) Zbl 1143.65384
Summary: An Eulerian-Lagrangian method of fundamental solutions (ELMFS) is developed by combining the Eulerian-Lagrangian method (ELM) and the method of fundamental solutions (MFS) to solve advection-diffusion problems. An advection-diffusion problem is first transformed into a diffusion problem using the ELM. Then the MFS is used to get the numerical solution as a linear combination of the fundamental solution of the diffusion operator. The ELMFS can handle not only constant advection velocity field but also variable advection velocity field. Following the properties of the MFS, the ELMFS is free from singularities, numerical integrations, and meshes.
Examples on advection-diffusion problems with varying propagation velocities for 2D and 3D cases are solved by the ELMFS and comparisons are carried out with the analytical solutions. The test results obtained for all the validation problems are in good agreements with the results available in the literature.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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