Adaptive Mesh Refinement Methods Numerical Solution of PDEs

Ravi Samtaney, Ph.D.

Princeton Plasma Physics Laboratory

In many science and engineering applications with multiple length and temporal scales, we compute systems of coupled nonlinear partial differential equations (PDEs). We frequently encounter localized regions in space where near-singular layers exist or where the solution develops discontinuities (for example, in nonlinear hyperbolic PDEs, the solutions are not guaranteed smoothness and can develop discontinuities such as shocks). Adaptive mesh refinement (AMR) methods provide one remedy to deal with multiple spatio-temporal scales by acting as a "numerical microscope", dynamically providing resolution in regions dominated by the discontinuities or singularities.

This talk will be divided into two broad segments. In the first segment, we will present a general overview of structured AMR methods for solving hyperbolic and elliptic PDEs. In the next segment, the focus will be on presenting illustrative examples from applications in the field of magneto-hydrodnamics (MHD) of relevance to the fusion program. These will include pellet injection , a proven method to refuel tokamaks; magnetic reconnection which is a canonical problem in plasma physics involving thin current sheets; and a example in MHD shock refraction where five or more discontinuities meet at a single point.

 

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