Obtain the Solution to the Poisson Equation Using the Finite Difference Method
The finite difference method is an approximate method used to solve a wide range of problems involving partial differential equations.
The finite difference method converts partial differential equations into a set of linear equations and solves them using matrix inversion.
Obtaining the solution to the Poisson equation using the finite difference method replaces continuous field problems having infinite degrees of freedom by a discretized field of finite regular modes.
The most practical and frequently used partial differential equation is the Poisson equation
In engineering, there are a variety of physical situations that engineers must manage. Most of these situations can be described using mathematical equations. One such equation is the Poisson equation, which governs physical situations such as diffusion, gravitation, and electrostatics. The Poisson equation can be solved using various numerical methods. Obtaining the solution to the Poisson equation with the finite difference method (FDM) is popular with engineers. In this article, we will explore the Poisson equation and the finite difference method further.
The Poisson Equation in Engineering
In engineering, mathematical modeling of physical phenomena is common. Most physical phenomena (when modeled mathematically) form partial differential equations (PDEs). The most practical and frequently used partial differential equation is the Poisson equation.
The Poisson equation is an elliptic partial differential equation that governs the mathematical modeling of electromagnetic, electrostatic, gravitational, and diffusion problems, to name a few. The finite difference method is an approximate method that is used to solve a wide range of problems involving partial differential equations. Problems can be time-independent, time-dependent, linear, or non-linear.
The finite difference method is suitable for solving problems with different kinds of boundary conditions such as Dirichlet, Neumann, etc. It is applicable to problem domains with different boundary shapes or regions composed of different materials.
Let’s look at a few examples of physical situations where the mathematical model leads to the Poisson equation.
Examples of Physical Phenomena Expressed With the Poisson Equation
Diffusion equation - In diffusion problems, the flux is represented in terms of the quantity of chemical solute and diffusivity (k). Steady-state diffusion can be described in the form of the Poisson equation as the following, where S(x) is the source of solute:
Heat diffusion equation - The heat diffusion equation is expressed in terms of possible sources of heat and the coefficient of thermal diffusivity. The equation is:
H is the heat field, T is the temperature field, K is a constant, and S(x) is the possible source of heat.
Electrostatic equation - From the laws of electrostatics, an equation relating the electric charge (P), electric field (E), and permittivity can be derived as:
Schrodinger’s equation in quantum mechanics.
The motion of biological organisms in a solution.
Even though the Poisson equation is a powerful tool for modeling engineering situations, the possibility of analytically solving the equation is applicable to only simple geometries. When modeling the behavior of systems with complex geometries, it is common to depend on numerical techniques. Several numerical simulations and competing algorithms are available for solving the Poisson equation. However, the finite difference method is the simplest method to use.
The Finite Difference Method
As the Poisson equation is solvable only for a handful of simple engineering models, computational algorithms are employed to achieve approximate numerical solutions. Among the numerical techniques, the finite difference method is the oldest, simplest, and most straightforward method to solve the Poisson equation.
The finite difference method converts partial differential equations into a set of linear equations and solves them using matrix inversion. In FDM, the partial differential equation is directly converted from a continuous function and operator into discretely-sampled counterparts. The accuracy of FDM is related to the capability of a finite grid to approximate the continuous function. The percentage of error in the solution can be minimized by increasing the sample numbers in FDM.
Solving the Poisson Equation With the Finite Difference Method
There are defined steps that must be taken when solving the Poisson equation with FDM. The partial differential equations are discretized, and the discretization can be carried out in two ways:
Discretization over a uniform grid - The mesh point is constant.
Discretization over a non-uniform grid - The distance of the mesh points is not constant.
Once the grids or mesh points are generated uniformly or non-uniformly, the Poisson equation is replaced by a finite difference approximation. The linear system of algebraic equations obtained after discretization is solved using direct or iterative methods. By solving a given grid or mesh point, an approximate solution of the Poisson equation that satisfies all the grid points is obtained.
Using FDM to solve the Poisson equation replaces continuous field problems having infinite degrees of freedom by a discretized field of finite regular modes. The finite difference method benefits the scientific and industrial community by offering a direct and intuitive approach to solving the Poisson equation. Simple coding and economic computation are the greatest of all benefits offered by FDM.
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