Forward Finite Differene Method

[roldy]

Homework Statement

For a given function u=u(x,t), start with Taylor series to implement the forward finite difference method to the heat conduction equation in non-dimensional form

$\frac{\partial u}{\partial t}=\frac{\partial ^2u}{\partial x^2}$

by using the following steps:

a) Obtain the first three truncated terms for each derivative in Eq. (1), using the method discussed in class. Your answer should be in terms of the step sizes $\Delta t$ and $\Delta x$. Use subscript i for x and superscript n for t to represent the nodes.

b) From your answers in Part a), determine the order of the truncation error in terms of the step sizes.

c) As an example, given the function

$u=(2^x)sin(t)$

In which the non-dimensional time t is in radians, calculate the exact value of the truncation error for $\partial u/ \partial t$ at node (i,n+1) where non-dimensional variables have values x=0.22100 and t=1.0472. Use $\Delta t=0.015$ and $\Delta x=0.22$.

d) Calculate the value of the leading truncated term and compare to the exact value of the truncation error.

Keep 5 significant digits in your numerical answers.

Homework Equations

$\frac{\partial u}{\partial t}=\frac{\partial ^2u}{\partial x^2}$

Taylor series:

$u(x_0+\Delta x)=u(x_0)+\frac{\Delta x}{1!}u'(x_0,u(x_0))+\frac{\Delta x^2}{2!}f''(x_0,u(x_0))+\frac{\Delta x^3}{3!}u'''(x_0,u(x_0))+…$

The Attempt at a Solution

I tried to solve this problem a couple times already but upon my last attempt I realized that I was not answering what the problem was asking for. I'm stuck on understanding the part a). Here's what I'm thinking…I'm suppose to take the derivative terms of the Taylor series and substitute in the forward finite difference form for each term. However, the $\Delta x$ terms will cancel. This does not seem correct. I'm not sure what the problem means when it says to start with the Taylor series to implement the forward finite difference method. Also, the first sentence of part a) had me confused. Could someone provide some clarification to this problem so that I may work on the other parts.

 

[mighty2000]

Thank you for your post. I can understand how the first sentence of part a) may be confusing. Let me try to clarify it for you.

In order to implement the forward finite difference method, we need to discretize the heat conduction equation in non-dimensional form. This means that we need to approximate the derivatives in the equation using finite differences.

The Taylor series expansion is a useful tool for approximating functions, and we can use it to approximate the derivatives in the heat conduction equation. In this case, we are using the Taylor series to approximate the derivatives at a specific node (i,n) in terms of the step sizes \Delta t and \Delta x.

So, in part a), you are essentially taking the derivatives of the Taylor series and substituting them into the heat conduction equation to obtain the truncated terms. The \Delta x terms may not cancel out completely, as they will depend on the specific terms in the Taylor series.

I hope this helps clarify the problem for you. Please let me know if you have any further questions. Good luck with your solution.