computational physics partial diferential equations
TRANSCRIPT
Partial Diferential Equations
Computational Physics
Partial Diferential Equations
Laplace's EquationWave Equation
Outline
Laplace's Equation
Finite Diference Equation
Solution Method
Boundary Conditions
Example
Wave Equation
Finite Diference Equation
Solution Method
Boundary Conditions
Example
Partial Diferential Equations
Laplace's Equation:
Wave Equation:
Difusion (Heat) Equation:
Grid
x
y
i i+1i-1
j-1
j+1
j
y Diference Equation
x
y
i i+1i-1
j-1
j+1
j
x Diference Equation
x
t
i i+1i-1
j-1
j+1
j
Full Diference Equation
Rearrange assuming square grid
y diference equation + x diference equation = 0
Central value is average of neighbor values
x
y
i i+1i-1
j-1
j+1
j
Boundary Conditions
x
y
boundary conditions
bou
nd
ary
con
dit i o
nsb
ou
nd
ary
con
dit
ion
s
boundary conditions
Method of Solution
We can derive V at a point in the gridfrom knowledge of its neighbors.
Boundary Conditions provide information forsome, but not all, neighbors.
Iterative Solution Method (Relaxation):
Assume initial, arbitrary, value for interiorpoints in grid
Fill in grid using diference equation
Repeat operation until convergence
ExampleTemperature of 2D Sheet
100
100100
0
Two DimensionalSheet with boundary T setto 100 on threesides and 0 onfourth side
Initial guess setsall points at 0
1
2
16
128
1024
8192
Convergence
ConvergenceRMS Results
N Mean Square1 34.65372 10.24934 3.17678 1.0351
16 0.346632 0.117764 0.0401
128 0.0137256 0.0046512 0.0016
1024 0.0004942048 0.00006434096 0.000001138192 0.000000000347
Partial Diferential Equations
Laplace's Equation:
Wave Equation:
Difusion (Heat) Equation:
Grid
x
t
i i+1i-1
j-1
j+1
j
t Diference Equation
x
t
i i+1i-1
j-1
j+1
j
x Diference Equation
x
t
i i+1i-1
j-1
j+1
j
Full Diference Equation
Rearrange
x diference equation = t diference equation
Alpha must be <1for stable solution
x
t
i i+1i-1
j-1
j+1
j
Boundary Conditions
x
t
Initial Conditions:Need V and dV/Dt
bou
nd
ar y
con
dit i o
nsb
ou
nd
ary
con
dit
ion
s
Initial Conditions
Initial Conditions require both V anddV/dt since time derivative is secondorder
Assuming dV/dt = 0:
Boundary ConditionsSpecify Value (Dirichlet)
x
t
2 3 1
j-1
j+1
j
Boundary Valuesupplied to f indsolution at next time step
Boundary ConditionsSpecify Derivative (Neumann)
x
t
2 3 1
j-1
j+1
j
Solution mustbe found on boundary.
Set value outside sol'ngrid to match derivative
Example
TIMEPosition