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