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.
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.
Read more about this topic: Discrete Poisson Equation