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Lecture 7

Lecture 7. Grids. Static Grids: Uniform Grids; Cartesian Curvilinear Non-Uniform Grids; Irregular Grids; Dynamic Grids: AMR (Adaptive Mesh Refinement). Grids. Grid Geometry Dependences Form of the Equations Numerical Differentiation Boundary Conditions Why? CPU time Memory

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Lecture 7

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  1. Lecture 7

  2. Grids Static Grids: Uniform Grids; • Cartesian • Curvilinear Non-Uniform Grids; Irregular Grids; Dynamic Grids: AMR (Adaptive Mesh Refinement)

  3. Grids Grid Geometry Dependences • Form of the Equations • Numerical Differentiation • Boundary Conditions Why? • CPU time • Memory • Accuracy

  4. Uniform Grid Numerical Differentiation Forward y’(x) = [y(x) – y(x-h)]/h Backward y’(x) = [y(x+h) – y(x)]/h Three point derivative

  5. Uniform: Polar Curvilinear coordinates 1.) Euler Equation: additional terms 2.) Curvilinear Derivatives

  6. Non-Uniform Grids Numerical Derivative: Stretching factor: limited

  7. Non-Uniform Curvilinear f/r assymmetry Radial stretching

  8. Chebyshev Grid Chebyshev polynomials Spectral Method on non-uniform grids

  9. Irregular Grids Grid Generation: Problem-specific grid geometry non-regular grids

  10. Complex boundaries varying cell geometry (industry)

  11. Irregular polar non-uniform

  12. AMR Adaptive Mesh Refinement: Multi-scale problems Scale 1: L1 ~ 0.01 m Scale 2: L2 ~ 100 m Dx ~ 0.001m L ~ 1000m N ~ 10^6 2D: N^2 ~ 10^12 3D: N^3 ~ 10^18

  13. AMR Dynamic mesh geometry: Adaptation to problem • Magnetic reconnection • Self gravity • Multiscale phenomena

  14. AMR problems Reconnection Gravitational clustering

  15. AMR Refinement levels

  16. AMR Star formation: Density structure of a barotropic collapse with magnetic field, computed with NIRVANA3. adaptive mesh refinement and self-gravitation. U. Ziegler (2005)

  17. AMR in action

  18. Summary Static Cartesian Grid (simple, fast) Polar, Spherical (rotation, axial symmetry) Non-Uniform (increasing resolution, static setup) Chebishev (Boundary effects, spectral) AMR (Multiscale problems) Direct comparison: More Complex, More CPU, Less Memory Performance = Balance (CPU,Memory)

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