Finite difference approximation question

Thread starter volcano90
Start date Sep 4, 2012
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Approximation Difference Finite Finite difference

In summary, a finite difference approximation is a common numerical method used to approximate solutions to mathematical problems by dividing them into smaller parts. It is often used in fields such as math, physics, and engineering and works by calculating the difference between neighboring points to approximate derivatives or integrals. However, it has limitations such as only providing an approximate solution and may not work well for certain types of problems. Other alternatives such as the finite element and boundary element methods may be more suitable for certain cases.

Sep 4, 2012

volcano90

Hi,

I have a question regarding finite difference approximation:
Consider the ﬁnite diﬀerence approximation
u'(xj-1/2) + λu(xj−1/2) ≈ 1/h*[u(xj ) − u(xj−1)] + λ(θu(xj ) + (1 − θ)u(xj−1))

how can I Find the order of approximation as a function of θ?

I am really new in this field, so appreciate any kind answer.

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Sep 5, 2012

volcano90

nobody knows the answer ?

1. What is a finite difference approximation?

A finite difference approximation is a numerical method used to approximate the solution to a mathematical problem by dividing the problem into smaller, simpler parts. It involves calculating the difference between values at neighboring points in the problem to approximate the derivative or integral of a function.

2. When is a finite difference approximation used?

Finite difference approximation is commonly used in fields such as mathematics, physics, and engineering to solve differential equations or other complex problems that cannot be solved analytically. It is also used in computer simulations and modeling.

3. How does a finite difference approximation work?

A finite difference approximation works by dividing the problem into a grid of points and using the values at these points to calculate the derivatives or integrals of the function. The smaller the grid size, the more accurate the approximation will be.

4. What are the limitations of finite difference approximation?

One limitation of finite difference approximation is that it can only provide an approximate solution, not an exact one. The accuracy of the approximation also depends on the grid size and the complexity of the problem. Additionally, it may not work well for problems with irregular boundaries or complex geometries.

5. Are there any alternatives to finite difference approximation?

Yes, there are other numerical methods such as finite element method and boundary element method that can also be used to approximate solutions to mathematical problems. These methods may be more suitable for certain types of problems and may provide more accurate results in some cases.