Discrete Poisson equation
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In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics.
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[edit] On a two-dimensional rectangular grid
Using the finite difference numerical method to discretize the 2 dimensional Poisson equation (assuming a uniform spatial discretization) on an m x n grid gives the following formula:
where and . The preferred arrangement of the solution vector is to use natural ordering which, prior to removing boundary elements, would look like:
This will result in an mn x mn linear system:
where
I is the m x m identity matrix, and D, also m x m , is given by:
[1] For each uij equation, the columns of D correspond to the u components:
while the columns of I to the left and right of D correspond to the u components:
and
respectively.
From the above, it can be inferred that there are n block columns of m in A. It is important to note that prescribed values of u (usually lying on the boundary) would have their corresponding elements removed from I and D. For the common case that all the nodes on the boundary are set, we have and , and the system would have the dimensions (m - 2) (n - 2) x (m - 2) (n - 2) , where D and I would have dimensions (m-2) x (m-2) .
[edit] Example
For a 5 x 5 ( m = 5 and n = 5 ) grid with all the boundary nodes prescribed, the system would look like:
with
and
As can be seen, the boundary u's are brought to the right-hand-side of the equation. [2] The entire system is 9 x 9 while D and I are 3 x 3 and given by:
and
[edit] Methods of Solution
Because is block tridiagonal and sparse, many methods of solution have been developed to optimally solve this linear system for . Among the methods are a generalized Thomas algorithm, cyclic reduction, successive overrelaxation, and Fourier transforms.
[edit] Applications
In Computational fluid dynamics, solving for the flow field of an incompressible fluid often requires a pressure corrector step to enforce the incompressibility. For such a fluid, the divergence of the velocity is zero, and the continuity equation is given as:
where vx is the velocity in the x direction and vy is velocity in y. Discretizing the continuity equation and making the proper substitutions gives a Poisson equation for the pressure in terms of the uncorrected velocity. [3].
[edit] Footnotes
- ^ Golub, Gene H. and C.F. Van Loan, Matrix Computations, 3rd Ed., The John Hopkins University Press, Baltimore, 1996, pages 177-180.
- ^ Cheny, Ward and David Kincaid, Numerical Mathematics and Computing 2nd Ed., Brooks/Cole Publishing Company, Pacific Grove, 1985, pages 443-448
- ^ Fletcher, Clive A. J., Computational Techniques for Fluid Dynamics: Vol I, 2nd Ed., Springer-Verlag, Berlin, 1991, page 334-339.
[edit] References
- Cheny, Ward and David Kincaid, Numerical Mathematics and Computing 2nd Ed., Brooks/Cole Publishing Company, Pacific Grove, 1985.
- Golub, Gene H. and C.F. Van Loan, Matrix Computations, 3rd Ed., The John Hopkins University Press, Baltimore, 1996.
- Hoffman, Joe D., Numerical Methods for Engineers and Scientists, 4th Ed., McGraw-Hill Inc., New York, 1992.
- Sweet, Roland A., SIAM Journal on Numerical Analysis, Vol. 11, No. 3 , June 1974, 506-520.