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Numerical Solution for one dimension linear wave problem

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Old   September 4, 2010, 06:27
Default Numerical Solution for one dimension linear wave problem
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I am a newbie in CFD.I need to solve the one dimension linear wave problem.I have found some source from wiki here and wikipedia.
http://www.cfd-online.com/Wiki/Linea...Runga-Kutta.29

I need to solve using 6th compaq scheme,1st order upwind scheme,2th maccormack scheme and lax-wendroff scheme.And compare the results of this different schemes.The time derivative needs to be solve using 4th runge-kutta method.Can anybody explain more on how to do the numerical solution?Thanks.And,how to integrate the 4th runge-kutta into the equation?
I need more help and resource on how to do it.
Attached Images
File Type: jpg 1.jpg (14.7 KB, 21 views)
File Type: jpg compact.jpg (33.2 KB, 18 views)
File Type: jpg upwind scheme.JPG (58.4 KB, 19 views)
File Type: jpg MacCormack.JPG (73.2 KB, 13 views)
File Type: jpg Lax–Wendroff method.JPG (56.2 KB, 15 views)

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Old   September 5, 2010, 07:33
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I have completed the numerical solution using maccormack scheme.Still need help for the other methods and on how to integrate the runge-kutta method into the time derivative.
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File Type: jpg print2.JPG (78.2 KB, 18 views)

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Old   September 7, 2010, 07:44
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Hello,

have you checked your solution with the exact solution of the wave equation ?

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Old   September 10, 2010, 04:30
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hello,Do.Thanks for your advice and support.
I have consulted my professor,the above graph is the exact solution since the cfl number is equal to one.
I have solve the equation using lax-wendroff,maccormack,and upwind scheme.
Now,I need to solve the wave equation using sixth-order compact scheme and time-integrated by a four-stage runge-kutta method.I really dont have any idea how to do it.Is there anybody know how to do it?I need some materials and guidance.Thanks!
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Old   September 10, 2010, 07:36
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Basically you need to "split" your problem in two parts:

1. Replace the spatial derivative with a 6th order finite difference formula and move this in the right hand term. You will have now a problem of this form:

df/dt=R(x)

where R(x) contains your discretized spatial terms.

2. For temporal discretization use any good book of numerical methods, or simply search Google for Runge-Kutta ...


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Old   September 10, 2010, 07:41
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About your professor confirmation that your solution is the "exact one" for CFL=1. This is true only if your implementation if faultless - this is exactly the point of comparing the "exact" analytical solution with the numerical one to prove that your implementation is correct.

From my experience the time spent in comparing a numerical solution with an analytical solution (when you have one) is always rewarded .

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Old   September 10, 2010, 08:23
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Thanks !
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Old   September 10, 2010, 22:10
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when I solve for the runge-kutta equation,I use R(x) as the slope?
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