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PDE’s in Curvilinear Coordinate Systems

M 693 b Advanced Numerical Analysis. PDE’s in Curvilinear Coordinate Systems. Rodrigo Negreiros – Ron Caplan – Joan Martinez. Overview. Grids in curvilinear coordinates Poisson Eq. in Polar Coordinates Wave Eq. in Polar Coordinates Heat Eq. in Polar Coordinates

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PDE’s in Curvilinear Coordinate Systems

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  1. M 693 b Advanced Numerical Analysis PDE’s in Curvilinear Coordinate Systems Rodrigo Negreiros – Ron Caplan – Joan Martinez

  2. Overview • Grids in curvilinear coordinates • Poisson Eq. in Polar Coordinates • Wave Eq. in Polar Coordinates • Heat Eq. in Polar Coordinates • Heat Eq. in Spherical Coordinates

  3. Grids in curvilinear coordinates

  4. Grids in curvilinear coordinatesAnother Approach

  5. Poisson Equation • Poisson Equation • Poisson Equation in Polar Coordinates

  6. Poisson Equation • Five point Laplacian • Conjugate Gradient Method • Round off errors ???

  7. Results

  8. 2D Heat Equation on a DiskUt = b(Uxx + Uyy) • New R=0 Approach • Different Stability Requirements R vs. Phi • Boundary Conditions • Not Symmetric

  9. Wave equation in a disk • The second order wave equation is given by • In polar coordinates last eq. becomes

  10. Drumhead

  11. Drumhead - results

  12. Drumhead - results

  13. Drumhead - results

  14. Drumhead - remarks • Method very stable for smooth initial conditions. • Extremely sensitive to non-smooth initial conditions: requires very fine grid.

  15. Heat equation on the surface of a sphere • Heat equation • On the surface of the sphere

  16. Results

  17. Results

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