Solving the algebraic equations
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Solving the algebraic equations. A x = B = =. 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).

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Solving the algebraic equations l.jpg

Solving the algebraicequations

A x = B

= =


Direct solution l.jpg
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)


Gauss elimination l.jpg

substitute in the 2nd equation

extract x2

Gauss elimination

Tridiagonal 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)


Iterative methods l.jpg

- Correct

from the value of

is small enough

- continue until

The method converges if

Iterative methods

Guess a solution


General iterative procedure l.jpg

pre-condition system

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 l.jpg
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 l.jpg
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


Multigrid methods 2 l.jpg
Multigrid methods (2)

long-range errors

and sampling

short-range errors

R

R

x

x

x

x

x


Decoupling the equations l.jpg
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 l.jpg
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 cont12 l.jpg
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 l.jpg
Fourier transform method

Consider the 2-dimensional Poisson equation in finite-differences

or

where

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


Fourier transform method cont l.jpg

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 l.jpg
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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