Is there any TVD scheme, which can give second order accuracy even at sonic points or in the region of large gradients. It will be a great help if somebody tell me.

regards Ritesh

I think TVD can't give high accuracy (> 1-st) at the discontinuity. But WENO scheme can do it.

See

Jiang, G.,Shu C., Efficient Implementation of Weighted ENO Scheme, // J. Comp. Phys., (126), 1996, pp. 202-228.

Alexey.

Why do you want second order accuracy at these points?

As a rule-of-thumb, if you have not used sufficient grid resolution to resolve gradients you will usually get worse solutions with high order schemes compared to low order schemes.

As another rule-of-thumb, what seems to work best in such regions are schemes which preserve relevant physical properties. Many schemes sacrifice one or more orders of accuracy (given the size of their computational molecule) even in resolved regions to gain better physical behaviour.

Near a discontinuity there is no better resolution than first order and TVD schemes do just that. Near a sonic point, the problem is not the TVD scheme but the underlying numerical scheme (Roe's for example). If you are concerned about the amount of numerical dissipation of TVD schems try WENO or MP-WENO schemes.

Thanks for replying, actually I just wanted to have second order accuracy of at sonic points. Now I know it is not possible with TVD scheme, Infact today I read that it is impossible to have second order accuracy using TVD scheme in the book "Numerical Methods for Conservation Laws" by R. J. Levaque. Any futher suggetions? Thanks once again.

regards Ritesh