Applying the Finite Difference Method in Electromagnetics to Solve Partial Differential Equations

Key Takeaways

• The finite difference approximation is the simplest numerical method to solve differential equations. 

• In the finite difference method, differential equations defined over a continuous region of three-dimensional space are replaced by a set of discrete equations, called finite difference equations.

• The finite difference method in electromagnetics is an excellent numerical method for solving partial differential equations when the solution region is of regular geometries.

Maxwell’s equations are important in solving electromagnetic field problems

Maxwell’s equations are critical to solving electromagnetic field problems. Electromagnetics is solely based on solving differential equations with boundary conditions and initial conditions. Obtaining solutions for partial differential equations involved with electromagnetics is a tedious task, especially when approaching it analytically. Numerical methods such as the finite element method, finite time-domain method, and finite difference method are used to solve partial differential equations in electromagnetic field problems. The application of the finite difference method in electromagnetics discretizes the continuous domain and converts differential equations into linear algebraic equations, which are easier to solve. In this article, we will take a closer look at this method and its advantages.

The Application of Numerical Methods in Electromagnetics

Any electromagnetic field that exists satisfies Maxwell’s equations and its boundary conditions. The differential form of Maxwell’s equations describes the variations in the fields and fluxes present in the three-dimensional space that is under consideration. A general solution to Maxwell’s equations can be obtained by using either analytical methods or numerical methods.

The numerical methods are based on approximating the fields and fluxes to the desired level of accuracy before solving them. The finite-difference approximation is the simplest numerical method to solve differential equations. Let’s discuss how this method works and the steps involved.

The Finite Difference Method in Electromagnetics

Using the finite difference numerical method in electromagnetics became popular with the advent of fast digital computers equipped with high memory space, and this method is now widely used. The finite difference method in electromagnetics is a numerical procedure based on approximations to solve partial differential equations. This method can be applied when solving linear, non-linear time-independent, and time-dependent problems. One of the main fields of application is electromagnetics, where the method can be used to solve problems pertaining to different shapes, boundary conditions, initial conditions, and regions involving non-homogenous materials. 

How the Finite Difference Method Works

The finite difference method in electromagnetics utilizes simple arithmetic operations to solve complex equations in differential form. The approach of solving partial differential equations using the finite difference method starts with discretization. In the finite difference method, differential equations defined over a continuous region of three-dimensional space are replaced by a set of discrete equations, called finite difference equations. The region of interest in the problem is defined as regular grids in the finite difference method, and this helps to achieve stable, accurate, and efficient solutions. 

The Steps Involved in the Finite Difference Method

There are three main steps involved in solving any partial differential equation using the finite difference method. They are:

1. Discretization of the solution region - This is the process of converting the solution region into a grid of nodes. The solution region is divided into meshes with grid points or nodes. The nodes in the boundary regions are called fixed nodes and the nodes in the interior regions are called free points. 

2. Approximating differential equations into finite difference equations - At finite discretization points, partial derivatives are approximated as a set of algebraic equations called finite difference equations. These difference equations relate the dependent variable at one node in the solution region to its neighboring nodes. 

3. Solving the difference equation - The set of finite difference simultaneous equations are subjected to boundary conditions or initial conditions, and a generalized solution for the problem is obtained. 

Among the numerical methods used to solve electromagnetic problems, the finite difference method is popular due to its simplicity. The finite difference method in electromagnetics is the suggested numerical method to solve partial differential equations when the solution region is of regular geometries.

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