Computational Methods

| December 19, 2015

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

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∇ 2u(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:

⎛ ∂ 2u(x, y,t) ∂ 2u(x, y,t) ⎞ ∂u(x, y,t)
10 −4 ⎜
+
⎟⎠ =
∂x 2
∂y 2
∂t

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for 0 ≤ x ≤ 4, 0 ≤ y ≤ 4 0 ≤ t ≤ 5000
u(x, y, 0) = 0
u(x, y,t) = e y cos x − e x cos y for x = 0, x = 4, y = 0, y = 4

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Present results for t= 5000

Problem #2: Crank-Nicolson Problem
Solve the Two-Dimensional Temperature Problem above using Crank-Nicolson Method.

ENGR516 Assignment #5: Hyperbolic PDEs
ENGR 516 Computational Methods for Graduate Students
Catholic University of America
Problem #1: Two Dimensional Hyperbolic
Equation

Assignment #8
Expand the explicit method for hyperbolic
equation to two dimensions and solve:
Two Dimensional Wave Vibration Over Square Membrane
⎛ ∂ 2u(x, y,t) ∂ 2u(x, y,t) ⎞ ∂ 2u(x, y,t)
0.25 ⎜
+
⎟⎠ =
∂x 2
∂y 2
∂t 2

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for 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 and 0 ≤ t ≤ 2
u(0, y,t) = 0, u(2, y,t) = 0
u(x, 0,t) = 0, u(x, 2,t) = 0
u(x, y, 0) = 0.1sin(π x)sin(π y / 2),

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∂u(x, y, 0)
=0
∂t

Present Results for t = 0.1 and t = 1.8

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