2nd order TVD scheme

Wenjun · March 13, 2013, 09:03

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.TVD scheme.png

order.doc

solution.pdf

cfdnewbie · March 13, 2013, 17:01

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....

Wenjun · March 13, 2013, 21:45

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.

Wenjun · March 13, 2013, 21:55

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?

cfdnewbie · March 14, 2013, 03:06

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!

k_tafazoli · March 14, 2013, 03:24

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 ?

Wenjun · March 14, 2013, 03:25

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

Wenjun · March 14, 2013, 03:51

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

k_tafazoli · March 14, 2013, 13:31

Thanks anyway

March 13, 2013, 09:03	2nd order TVD scheme	#1
Wenjun New Member Join Date: Mar 2013 Posts: 14 Rep Power: 13	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.TVD scheme.png order.doc solution.pdf Baka BOi likes this.

March 13, 2013, 17:01		#2
cfdnewbie Senior Member cfdnewbie Join Date: Mar 2010 Posts: 557 Rep Power: 20	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.... Wenjun likes this.

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March 13, 2013, 21:45		#3
Wenjun New Member Join Date: Mar 2013 Posts: 14 Rep Power: 13	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.

March 13, 2013, 21:55		#4
Wenjun New Member Join Date: Mar 2013 Posts: 14 Rep Power: 13	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?

March 14, 2013, 03:06		#5
cfdnewbie Senior Member cfdnewbie Join Date: Mar 2010 Posts: 557 Rep Power: 20	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!

March 14, 2013, 03:24		#6
k_tafazoli New Member kian Join Date: Feb 2013 Posts: 14 Rep Power: 13	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 ?

March 14, 2013, 03:25		#7
Wenjun New Member Join Date: Mar 2013 Posts: 14 Rep Power: 13	Yes, I understand now. I tried L1, it works. Thank you very much. It is my pleasure to meet you in the forum.

March 14, 2013, 03:51		#8
Wenjun New Member Join Date: Mar 2013 Posts: 14 Rep Power: 13	Sorry, I only code by Matlab. I am a fresh guy in CFD and my problem is rather simple.

March 14, 2013, 13:31		#9
k_tafazoli New Member kian Join Date: Feb 2013 Posts: 14 Rep Power: 13	Thanks anyway