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 gridEdit

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).

ExampleEdit

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 solutionEdit

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]

 
Poisson convergence of various iterative methods with infinity norms of residuals against iteration count and computer time.

ApplicationsEdit

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]

FootnotesEdit

  1. ^ Hoffman, Joe (2001), "Chapter 9. Elliptic partial differential equations", Numerical Methods for Engineers and Scientists (2nd ed.), McGraw–Hill, ISBN 0-8247-0443-6.
  2. ^ Golub, Gene H. and C. F. Van Loan, Matrix Computations, 3rd Ed., The Johns Hopkins University Press, Baltimore, 1996, pages 177–180.
  3. ^ Cheny, Ward and David Kincaid, Numerical Mathematics and Computing 2nd Ed., Brooks/Cole Publishing Company, Pacific Grove, 1985, pages 443–448.
  4. ^ CS267: Notes for Lectures 15 and 16, Mar 5 and 7, 1996, https://people.eecs.berkeley.edu/~demmel/cs267/lecture24/lecture24.html
  5. ^ Fletcher, Clive A. J., Computational Techniques for Fluid Dynamics: Vol I, 2nd Ed., Springer-Verlag, Berlin, 1991, page 334–339.
  6. ^ S. P. Meyn and R.L. Tweedie, 2005. Markov Chains and Stochastic Stability. Second edition to appear, Cambridge University Press, 2009.
  7. ^ S. P. Meyn, 2007. Control Techniques for Complex Networks, Cambridge University Press, 2007.
  8. ^ Asmussen, Søren, Glynn, Peter W., 2007. "Stochastic Simulation: Algorithms and Analysis". Springer. Series: Stochastic Modelling and Applied Probability, Vol. 57, 2007.

ReferencesEdit

  • 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.