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Enhanced Ideal MHD Solver via Unsplit Godunov Method

Explore the application of Corner Transport Upwind, Upwinded Constrained Transport, and modifications to PPM scheme for multidimensional ideal MHD simulations. Enhance characteristic tracing and sweeping across a 3D grid. Utilize CTU scheme and HD Stencil for improved edge updating and field evolution equations. Incorporate constraints on ∇•B to prevent erroneous growth in 2D and 3D. Optimize multidimensional source terms to reduce to 2D scheme for grid-aligned flow.

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Enhanced Ideal MHD Solver via Unsplit Godunov Method

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  1. An unsplitgudonov method for ideal MHD via Constrained Transport in Three Dimensions Gardiner and Stone

  2. Outline • Corner Transport Upwind of Colella (CTU) • Upwinded Constrained Transport (UCT) • Modifications to Characteristic Tracing of PPM scheme. • Sweeping across a 3D grid.

  3. Corner Transport Upwind • CTU scheme of Colella • Use first set of Riemann solves to update perpendicular edges before second set of Riemann solves.

  4. HD Stencil

  5. HD Stencil

  6. HD Stencil

  7. HD Stencil

  8. Upwinded Constrained Transport • Ec CT algorithm of Gardiner and Stone • Use cell centered E as well as face (edge) centered E to construct edge (corner) centered E in an upwinded fashion. • Interpolate from cell facesto cell edge using gradientcomputed by differencingthe cell center and cell facein the upwind direction.

  9. MHD Stencil

  10. MHD Stencil

  11. MHD Stencil

  12. MHD Stencil

  13. MHD Stencil

  14. MHD Stencil

  15. Multi-dimensional modifications to PPM Characteristic Evolution • The piece-wise parabolic method of Colella is a dimensionally split method and involves a one-dimensional parabolic spatial reconstruction and characteristic evolution of the primitive variables to get the time averaged edge values where the characteristic evolution step is calculated by solving • Where and

  16. In 2D • The resulting evolution equation for the magnetic field from the one-dimensional characteristic tracing is: • It is however missing some terms compared to the split form of the induction equation in 2D: • In 2D, the constraint on B can be incorporated into the split form of the evolution equation for Bz. This helps prevent erroneous growth in Bz for grid-aligned flow.

  17. In 3D • In 3D, it is not clear how to use the constraint on B to modify the split form of the induction equation. The goal is to allow for multidimensional source terms, but in a way that reduces to the 2D scheme for grid-aligned flow. • If xBxand yByhave the same sign, then the source terms reduce to the split form of the induction equation. • If they are equal and opposite however (as they will be in 2D since zBz= 0), then the evolution equation for Bz reduces to the 2D form.

  18. Sweeping the grid • First determine overall range of dependence for different stencil pieces: • Second, determine lag and lead for different stencil pieces.

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