Numerical Solution for one dimension linear wave problem
 

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.

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.

Hello,

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

Do

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!

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

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

Thanks !

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