# Computational Methods

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

!

!

!

∇ 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

⎝

!

!

!

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

!

!

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

⎝

!

!

!

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),

!

!

∂u(x, y, 0)

=0

∂t

Present Results for t = 0.1 and t = 1.8

**Category**: Essay