Discrete Poisson equation
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.
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 × n grid gives the following formula:[1]
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 × mn linear system:
where
is the m × m identity matrix, and , also m × m, is given by:
[2] and is defined by
For each equation, the columns of correspond to a block of components in :
while the columns of to the left and right of each correspond to other blocks of components within :
and
respectively.
From the above, it can be inferred that there are block columns of in . It is important to note that prescribed values of (usually lying on the boundary) would have their corresponding elements removed from and . 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) × (m − 2)(n − 2), where and would have dimensions (m − 2) × (m − 2).
Example
For a 5×5 ( and ) grid with all the boundary nodes prescribed, the system would look like:
with
and
As can be seen, the boundary 's are brought to the right-hand-side of the equation.[3] The entire system is 9 × 9 while and are 3 × 3 and given by:
and
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 with a resulting computational complexity of , cyclic reduction, successive overrelaxation that has a complexity of , and Fast Fourier transforms which is . An optimal solution can also be computed using multigrid methods. [4]
Applications
In computational fluid dynamics, for the solution of an incompressible flow problem, the incompressibility condition acts as a constraint for the pressure. There is no explicit form available for pressure in this case due to a strong coupling of the velocity and pressure fields. In this condition, by taking the divergence of all terms in the momentum equation, one obtains the pressure poisson equation.
For an incompressible flow this constraint is given by:
where is the velocity in the direction, is velocity in and is the velocity in the direction. Taking divergence of the momentum equation and using the incompressibility constraint, the pressure poisson equation is formed given by:
where is the kinematic viscosity of the fluid and is the velocity vector.[5]
The discrete Poisson's equation arises in the theory of Markov chains. It appears as the relative value function for the dynamic programming equation in a Markov decision process, and as the control variate for application in simulation variance reduction.[6][7][8]
Footnotes
- Hoffman, Joe (2001), "Chapter 9. Elliptic partial differential equations", Numerical Methods for Engineers and Scientists (2nd ed.), McGraw–Hill, ISBN 0-8247-0443-6.
- Golub, Gene H. and C. F. Van Loan, Matrix Computations, 3rd Ed., The Johns 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.
- CS267: Notes for Lectures 15 and 16, Mar 5 and 7, 1996, https://people.eecs.berkeley.edu/~demmel/cs267/lecture24/lecture24.html
- Fletcher, Clive A. J., Computational Techniques for Fluid Dynamics: Vol I, 2nd Ed., Springer-Verlag, Berlin, 1991, page 334–339.
- S. P. Meyn and R.L. Tweedie, 2005. Markov Chains and Stochastic Stability. Second edition to appear, Cambridge University Press, 2009.
- S. P. Meyn, 2007. Control Techniques for Complex Networks, Cambridge University Press, 2007.
- Asmussen, Søren, Glynn, Peter W., 2007. "Stochastic Simulation: Algorithms and Analysis". Springer. Series: Stochastic Modelling and Applied Probability, Vol. 57, 2007.
References
- 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.
- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 20.4. Fourier and Cyclic Reduction Methods". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.