Measuring Error in Finite Difference Electrostatic Solutions | Tips & Tricks

Other potential measures of error, such as relative and absolute error, may not be suitable for this problem due to the wide range of values involved. Additionally, the author is looking for any previous work or ideas on solving an electrostatic problem involving a dipole and concentric multi-layered spheres.
  • #1
Zhivago
26
1
Hello everyone

I am solving an electrostatic problem (with analytical solution) using a finite difference method and I want to plot the error of the FD against the analytical solution.
What would be a good way to measure the error?
I tried the relative and absolute error, but since values range from 0 to high values, these aren't a good measure in all regions.
I'm finding the potential on the outer surface of a group of concentric multi layered spheres caused by a dipole at an arbitrary position inside the inner sphere.
If anyone can point me to the work someone has done before, that would be good enough! Comments and ideas welcome.

Thank you for reading
 
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  • #2
.A good way to measure the error between the finite difference solution and the analytical solution is the root mean square (RMS) error. This compares the difference between the two solutions at all points in the domain and gives an overall measure of the accuracy of the finite difference approach. This is calculated by taking the square root of the sum of the squared differences between the two solutions at each discrete point in the domain.
 

Related to Measuring Error in Finite Difference Electrostatic Solutions | Tips & Tricks

1. What is error analysis in FD?

Error analysis in FD, or finite difference, is a method commonly used in numerical analysis to approximate the derivatives of a function. It involves calculating the difference between values of a function at different points in a given interval and using this information to estimate the derivative.

2. Why is error analysis important in FD?

Error analysis is important in FD because it allows us to understand the accuracy and reliability of our approximations. By analyzing the errors involved in our calculations, we can determine how close our results are to the true values and identify any sources of error that may need to be addressed.

3. What are the main sources of error in FD?

The main sources of error in FD include truncation error, round-off error, and discretization error. Truncation error refers to the error introduced by using a finite number of terms in a series approximation. Round-off error is caused by the limited precision of numerical calculations. Discretization error arises from approximating a continuous function using a finite number of discrete points.

4. How is error analysis performed in FD?

Error analysis in FD is typically performed by calculating the error bound, which is a theoretical limit on the maximum possible error for a given method. This is done by using mathematical formulas and techniques such as Taylor series expansions and the Mean Value Theorem to analyze the behavior of the function and its derivatives.

5. What are some ways to minimize error in FD?

To minimize error in FD, one can use methods such as increasing the number of points used in the approximation, using higher order methods, and reducing the step size. It is also important to carefully choose the function and interval of interest to ensure that the approximation is as accurate as possible. Additionally, avoiding division by small numbers and using efficient algorithms can help reduce errors in FD.

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