ENGR 516 Computational Methods for Graduate Students

ENGR 516 Computational Methods for Graduate Students
Catholic University of America
Assignment #7
Elliptic PDE
ENGR516 Assignment #5: Two Dimensional PDEs
Problem #1: Alternating Direction Implicit (ADI) Method
Research and write a brief explanation of the ADI Method and solve:
Two Dimensional LaPlace Equation: Electric Potential Over a Flat Plate with Point
Charge
!!!
?2
u(x, y) = f (x, y) for -1 = x = 1, -1 = y = 1
boundary conditions: u(x,y) = 0 for all boundaries
f (0.5,0.5) = -1
f (-0.5,-0.5) = 1
elsewhere : f (x, y) = 0
Two Dimensional Temperature Diffusion:
!!!
10-4 ?2
u(x, y,t)
?x2 +
?2
u(x, y,t)
?y2
?
?
?
?
?
? = ?u(x, y,t)
?t
for 0 = x = 4, 0 = y = 4 0 = t = 5000
u(x, y,0) = 0
u(x, y,t) = ey cos x – ex cos y for x = 0, x = 4, y = 0, y = 4
! ! Present results for t= 5000
Problem #2: Crank-Nicolson Problem
Solve the Two-Dimensional Temperature Problem above using Crank-Nicolson Method.