Numerical Solution for one dimension linear wave problem

cxcxcx0505 · September 4, 2010, 07:27

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.

cxcxcx0505 · September 5, 2010, 08:33

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.

DoHander · September 7, 2010, 08:44

Hello,

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

Do

cxcxcx0505 · September 10, 2010, 05:30

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!

DoHander · September 10, 2010, 08:36

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 ...

Do

DoHander · September 10, 2010, 08:41

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

.

Do

cxcxcx0505 · September 10, 2010, 09:23

Thanks !

cxcxcx0505 · September 10, 2010, 23:10

when I solve for the runge-kutta equation,I use R(x) as the slope?

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September 7, 2010, 08:44		#3
DoHander Senior Member Join Date: Nov 2009 Posts: 411 Rep Power: 20	Hello, have you checked your solution with the exact solution of the wave equation ? Do

September 10, 2010, 05:30		#4
cxcxcx0505 New Member cxcxcx0505 Join Date: May 2010 Posts: 17 Rep Power: 16	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!

September 10, 2010, 08:36		#5
DoHander Senior Member Join Date: Nov 2009 Posts: 411 Rep Power: 20	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 ... Do

September 10, 2010, 08:41		#6
DoHander Senior Member Join Date: Nov 2009 Posts: 411 Rep Power: 20	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 . Do

September 10, 2010, 09:23		#7
cxcxcx0505 New Member cxcxcx0505 Join Date: May 2010 Posts: 17 Rep Power: 16	Thanks !