2nd order TVD scheme
 

Hi, I am learning the second order TVD scheme which is listed in the attachment, and I want to verify the order of the scheme by the linear advection equation u_t+u_x=0, periodic BC, initial condition u0=sin(pi*x), x in [0, 2]. But the order I got is about 1.3, and the wave shape becomes flat at valley and peak. Why is the order less than 2 ? My code is in the attachment too. Thank you very much for your kind reply.Attachment 19828

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Attachment 19830

looks like your TVD limiter works as expected and cuts off your peaks when a grid point coincides with the maxima/minina.
Your initial projection looks ok, you should approach second order when refining the mesh. Order of convergence is a concept of h to zero, from your plot it looks like h is 2/6....

Thank you for your consideration. But even if I choose the step h=2/6,2/12..., the order is almost the same as before. I don't know why. I have already checked my code for several times.

I use the L1 error for order calculation, it confirms the second order accuracy. However, L_\infty does not work. Normally the two different measure both can confirm the order. Do you have some idea about this problem?

The L infty will always be dominated by the limiter at the maxima, so since you are doing a first order approximation at the maxima (see your plot, the maxima are approximated by constants, not linears), the L infty will not give you second order. The L1 and L2 will converge to second order, since the influence of the "bad region" will decrease with increasing h!

Hi, I've got a fortran code for solving flow over an airfoil with the method of first order van leer flux vector splitting . I have to change it to second order . Do you know how I can do that ? can you help me with it ?

Yes, I understand now. I tried L1, it works. Thank you very much. It is my pleasure to meet you in the forum.

Sorry, I only code by Matlab. I am a fresh guy in CFD and my problem is rather simple.

Thanks anyway