Finite Difference Methods - Partial Differential Equations

1
Lecture 12

• Finite Difference Methods - Partial Differential
  Equations

2
Finite difference replacement (CTP3.1)

• x-? x x?

3
(No Transcript)
4

5
Finite difference replacement (CTP3.1)

6
(No Transcript)
7
Finite Difference Methods

• Partial differential equations
• Elliptic equations, boundary value problems

In studying Molecular Dynamics we were concerned
with an initial value problem, i.e. the function
and its derivative were given at one point
(t0) A one-dimensional boundary value problem
would be Schrodingers equation, with ?(x0) and
?(xL) given.
The methods for solution in the two cases are
quite different.
8
Example Elliptical equation, boundary value
problem
9
Procedure

• Construct a grid
• At all interior points replace the pde by finite
  difference version
• Solve these finite difference equations for the
  unknown values of the function

10
Grid (n1)(m1) points, hX/m, kY/n
11
At each interior point replace the p.d.e.
(?(xi,yj)??i,j)
Consider ?i,,j as an (n-1)(m-1) component vector
?, the equations take the form M?S, where S is a
column vector of N(n-1)(m-1) known quantities.
12
The exact solution ?ij of these equations is not
the same as the solution of the differential
equation F(xi,yj)
Truncation error ??ij-?(xi,yj) ?
13
With hk,
Case where f0,
14
Matrix M
15
Case Poissons equation, f0, with hk

• Solution exact solution ? exists
• Direct, e.g. ? M-1 S

Random Walk
Relaxation
16
Solution of the equations by relaxation
? satisfies
General linear iteration,
17
Guess a solution u(0)
initial error array e(0) ?- u(0)
Improved (?) solution
18
Error array after k iterations
19
where ?(J) is the spectral radius of J, the
largest in magnitude of its eigenvalues
20
But how quickly does the iteration converge?
21
Rate of convergence
But we dont know e(k)
Test for convergence