Difference between Central Difference Method and Finite Difference Method

[iqjump123]

Hello all,

I am in the process of solving a finite elements problem involving obtaining deflection of a simple mass-spring-damper 2nd order ODE system with a defined forcing function. While going through my class notes, I came across the idea of the central difference method, which is defining the derivatives of functions as functions by a certain time interval.

I was researching into how to solve this problem, but I noticed that the majority cited "central difference method" as just the function itself, not an actual "method", involving procedures. However, I saw informations involving solving ODEs through a "finite difference method", which looked like it used equations derived from the central difference method.

I was wondering what is the exact difference between the two terms, and if it makes sense that finite difference method is a method using the equations from the central difference method?

Thanks.

 

[AlephZero]

"Finite difference method" just means the general idea of approximating a function using a grid of points in space and/or time, and approximating derivatives from the function values at nearby points. The word "difference" comes from the basic idea of a Taylor series expansion: $y(h) - y(0) = h dy/dx$

There are many different ways to use that idea to solve different ordinary and partial differential equations. The "central difference method" is one way to solve 2nd order equations, like Newton's laws of motion.

 

[iqjump123]

Thanks for your reply!

I have actually posted another thread in a separate section about my problem that relates to this (cannot find link on mobile version at the moment)- in brief i need to find the deflections of a simple mass spring damper system in a truss evaluated with a forcing function using cdm. One issue that i simply cannot grasp is 1. how to solve this equation at each time point when the parameters ofthe differential equation is in matrices
2. My past skills involving matrices only involved a ax=b scheme. How would inverse matrices and cdm work to solve 2nd order odes??

I have looked at textbooks and schaums etc to get the answer but couldnt- maybe i just need a more specific answer, or just a nudge out the door- so to say- so that i can solve this issue.

Thanks again!

Iqjump123

 

[iqjump123]

By the way the link to my question is
https://www.physicsforums.com/showthread.php?t=556528

 

[blue_raver22]

The central difference method and finite difference method are both numerical methods used to solve differential equations, such as the mass-spring-damper system you are working on. The main difference between the two is the way in which they approximate the derivatives in the equations.

The central difference method involves approximating the derivative at a point by taking the average of the derivative at the points on either side of it. This is done by dividing the time interval into smaller intervals and using the values of the function at those points to calculate the derivative. This method is considered more accurate than other numerical methods, such as forward or backward difference methods, as it takes into account both the previous and future values of the function.

On the other hand, the finite difference method involves approximating the derivatives using finite differences, which are derived from the central difference method. This method is often used in solving partial differential equations, where the central difference method may not be applicable. It involves discretizing the domain into smaller elements and approximating the derivatives at each element using the values of the function at the element boundaries. This method is also known as the finite element method and is commonly used in engineering and scientific applications.

In summary, the central difference method is a specific way of approximating derivatives, while the finite difference method is a broader term that encompasses different methods using finite differences. It is correct to say that the finite difference method uses equations derived from the central difference method, but it also includes other techniques and approaches. Both methods have their own advantages and limitations, and the choice between them depends on the specific problem being solved.