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Direct solution

x = A-1 B

= =

• Applicable only to small problems

• For the vertical in the spectral technique where x is

a one-column vector

(decoupled equations in the horizontal)

substitute in the 2nd equation

extract x2

Gauss eliminationTridiagonal matrices: Large one-dimensional problems

substitute in the 3rd equation …. and so on

solve and substitute in the (n-1)th eq.

we arrive at a single equation for xn

solve for xn-1 and substitute in the (n-2)th eq. etc ……..

Pivots: a11 , a22-a21/a11 , … not too small (might need to rearrange order)

from the value of

is small enough

- continue until

The method converges if

Iterative methodsGuess a solution

add and substract

*

if

is the true solution

continuous equivalent of *

the general solution of this equation is:

where the λ’s are the eigenvalues of matrix

General iterative procedure• Convergence

it approaches the stationary solution k if Re(λ) < 0 (elliptic problem)

Example of iterative procedure

Helmholtz equation in finite differences

we have taken Δx=1 for simplicity

then

take

*

where

means all x from iteration nexcept xi,j from iteration n+1

this is the Jacobi method

if we take xi-1,j and xi,j-1 from iteration n+1, we have the Gauss-Seidel method

multiplying the correction in * by a factor μ>1, we have the overrelaxation method

Multigrid methods

- An iterative scheme is slow if the corrections from the initial guess are long-range corrections but very fast if they are local
- Multigrid methods first relax on a subset of the grid(therefore long-range corrections cover a lesser number of grid-points and are seen as more local)and then refine, relaxing on the original grid(or an intermediate one …) and the switching between grids is iterated
- This procedure is much more efficient than the straightforward relaxation and can compete with direct methods
- It is even more efficient in multiprocessor machines
- Adaptive multigrid methods only refine in the areas where the error is larger than a given threshold

Decoupling the equations

Assume we have a 3-D problem

tensor product

• Simplest case

that is

auxiliary vectors

solve

for each (m,n). Then solve

for each (i,n). Finally solve

for each (i,j).

Total O(I.J.K)3 operations

Decoupling the equations (cont)

• Use of the eigenvector matrix

Consider the Poisson equation in 3 dimensions

Using centered finite diff. In the vertical:

where

Is a matrix of rank K (No of levels)

Decoupling the equations (cont)

Let

be the eigenvectors of

calling

the matrix formed by the eigenvectors

being the diagonal matrix of eigenvalues

The discretized equation can then be written as:

K decoupled

equations

projections of φ along the eigenvectors

Fourier transform method

Consider the 2-dimensional Poisson equation in finite-differences

or

where

here Un: grid-point values of U in row n

The same holds for any other matrix of the form

(Helmholtz equation)

Fourier transform method (cont)A is a tridiagonal symmetric matrix whose eigenvalues are

j=1, 2, …, M

and the eigenvectors

are the Fourier basis functions

Fourier transform method (cont2)

Call

and

The original system may be written as follows:

Discrete Fourier transform

of vector of grid-points

at row k+1

decoupled system of equations for

the Fourier components (k=row number)

The projection to Fourier space and back can be done by FFT

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