computational physics partial diferential equations

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