# Finite Differences in Inhomogeneous Media

• Excom
In summary, the conversation discusses solving the Poisson equation using the Finite Difference Method for inhomogeneous media with embedded charge distributions. The person asks for literature recommendations on this subject. Suggestions include looking into Maxwell's Equations and texts such as Chew's Waves and Fields in Inhomogeneous Media or Jin's The Finite Element Method in Electromagnetics. Other important topics to consider are stability conditions and techniques for handling absorbing boundary conditions. Another possible method for solving Poisson's equation is the Method of Moments, but this may become more complicated with the addition of inhomogeneous media.

#### Excom

Hi

I am trying to solve the Poisson equation, with the use of the Finite Difference Method, for a inhomogeneous media with some charge distributions embedded in the media.

Is there anyone that know some literature, which treats this subject?

Not Poisson's specifically but you can find a bevy of material on doing Maxwell's Equations. The seminal paper is the Yee algorithm but you can find discussions about FDTD in Chew's Waves and Fields in Inhomogeneous Media or Jin's The Finite Element Method in Electromagnetics (not the best books on the subject but I can't remember the third text I am thinking of, author's name starts with a "T"). Probably the main thing that you need to learn is defining the stability conditions and looking into absorbing boundary conditions or perfectly matched layers though the latter is negated by sufficiently increasing the problem space.

EDIT: Taflove! That's his name. He has a great textbook all about FDTD. If you want to take a look at how to solve Poisson's equation using another technique, Harrington's text discusses how to solve it using the Method of Moments but adding inhomogenous medium makes it a little more annoying.

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for any help.

I am familiar with the use of the Finite Difference Method for solving the Poisson equation in various media. Inhomogeneous media, where charge distributions are embedded, can be challenging to solve using this method. However, there are several literature resources available that discuss this topic. Some examples include "Finite Difference Methods for Partial Differential Equations in Inhomogeneous Media" by John Strain and "Finite Difference Methods for Electromagnetics in Inhomogeneous Media" by Weng Cho Chew. Additionally, there are also many research papers published on this subject that may be helpful. I recommend consulting these resources to gain a better understanding of how to apply the Finite Difference Method in inhomogeneous media.

## 1. What are finite differences in inhomogeneous media?

Finite differences in inhomogeneous media is a numerical method used to approximate solutions to differential equations in media that have varying properties or parameters. It involves dividing a continuous domain into discrete points and approximating the derivatives at those points using finite difference equations.

## 2. How does finite differences in inhomogeneous media differ from finite differences in homogeneous media?

In finite differences in homogeneous media, the properties or parameters of the media are assumed to be constant throughout the domain. However, in inhomogeneous media, these properties or parameters can vary at different points in the domain, making the numerical method more complex.

## 3. What are some applications of finite differences in inhomogeneous media?

This numerical method is commonly used in fields such as physics, engineering, and geosciences to solve differential equations that describe phenomena in inhomogeneous media. Some examples include heat transfer in materials with varying thermal conductivity, wave propagation in heterogeneous media, and groundwater flow in porous media with varying permeability.

## 4. What are the advantages of using finite differences in inhomogeneous media?

One advantage is that this method can handle complex and non-uniform geometries and media properties. It is also relatively easy to implement and can provide accurate solutions with a relatively small number of grid points. Additionally, it can handle time-dependent problems and can be easily adapted to different types of boundary conditions.

## 5. What are some limitations of finite differences in inhomogeneous media?

One limitation is that the accuracy of the solutions depends on the choice of grid points and the size of the grid. In complex problems, a large number of grid points may be required, leading to longer computation times. Additionally, this method may not be suitable for problems with discontinuous media properties or steep variations in the properties.