pde’s in curvilinear coordinate systems rodrigo negreiros – ron caplan – joan martinez
TRANSCRIPT
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
• Heat Eq. in Spherical Coordinates
Grids in curvilinear coordinates
/ 2r r/ 2r
Grids in curvilinear coordinatesAnother Approach
r
r
Poisson Equation
• Poisson Equation
),(),(2 yxfyxu
• Poisson Equation in Polar Coordinates
)2
2
211 f(r,θr
θu
rru
rr
Poisson Equation
• Five point Laplacian
• Conjugate Gradient Method
• Round off errors ???
Results
2D Heat Equation on a DiskUt = b(Uxx + Uyy)
• New R=0 Approach• Different Stability Requirements R vs. Phi
• Boundary Conditions• Not Symmetric
• The second order wave equation is given by
• In polar coordinates last eq. becomes
Wave equation in a disk
22
2 2
10
t c
2 2 2
2 2 2 2
1 1
t r r r r
( 1)c
Drumhead 0R
Drumhead - results
Drumhead - results
Drumhead - results
Drumhead - remarks
• Method very stable for smooth initial conditions.
• Extremely sensitive to non-smooth initial conditions: requires very fine grid.
Heat equation on the surface of a sphere
• Heat equation
• On the surface of the sphere
2Tk T
t
2 2
2 2 2 2 2
1 1 cot
sin
T T T T
t R R R
Results
Results