I have been experimenting with using TVD schemes to solve the scalar convection equation given by dphi/dt + U*dphi/dx = 0. I am using 2nd Order Upwind differencing as higher order convection sheme. I am using explicit differencing for temporal discretisation. Eventually I plan to also test Crank-Nicholson and implict. So far I have made the following observations and would appreciate comments/answers.
(1) For initial condition that contain sharp discontinuities given by a square wave, the TVD schemes perform, if I may use a subjective description, quite well.
(2) For initial condition that is relatively smooth, the TVD schemes tend to sharpen the solution until it almost looks like a square wave, though it does not create any new mins or maxes. It seems for this case that it sharpens the profile, in contrast to first order upwinds drastic smearing.
(3) Observation (2) shows a courant number dependancy. For lower courant numbers the TVD scheme does not sharpen the profile as much.
(4)Are TVD schemes not valid for problems that do not have sharp discontinuities?
(5)If so does anyone have a suggestion for a general-purpose 2nd Order scheme that has flux limiting capabilities?
Re: TVD Schemes
You have to be a bit careful here! It is true that some TVD schemes sharpen the profile of the scalar if they introduce a certain level of Downwinding. The examples here would be SUPERBEE and van Leer. However, that is NOT the effect you're seeing. If you want to prove this to yourself, you can pick a 2-D test case (like a convection of a profile across a square domain - there is a nice selection of profiles in Leonard's paper). This test case is steady and you will see exactly what one can expect from TVD schemes.
What you are seeing is an effect of temporal discretisation. A while back I have managed to derive that explicit time-scheme introduces a negative numerical diffusion (sharpening of the profile) similar and opposite in effect to numerical diffusion of Upwind Differencing (UD). In fact, in 1-D with Co=1 the two effects exactly cancel out and you get the right solution with two first-order schemes!
You'll get the opposite effect with implicit temporal discretisation (positive numerical diffusion = smearing of the profile), whereas Crank-Nicholson is diffusion-neutral, but also unbounded (depending on exactly which TVD scheme you choose).
If you want the gory details (e.g. derivation of all this), please send me an E-mail and I'll pass it on to you.
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