PDE’s in Curvilinear Coordinate Systems

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

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

15. Drumhead

16. Drumhead - results

17. Drumhead - results

18. Drumhead - results

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

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

21. Results

22. Results