[TurbJet]

Originally Posted by FMDenaro

I wrote few lines about your problem, I suggest to read 16.4.1 and 16.4.2 in the book of Leveque. You will see that the definition of the upwind criterion is quite more complex than you supposed.

I looked up those two sections you suggested, what I get from that is, due to the spatially varying flux, at some point the equation will lose its hyperbolicity, which might leads to multiple waves involved in its solution. I guess it might cause the upwind somehow invalid.

I tried to consider negative velocity as well to my scheme, not just assume always positive. The solution does not blow up any more, but the square wave does not hold any longer: it only leaves a peak (or compressed wave) plus some small waves moving to the right.

Would it be safe to say that general upwind is not suitable for this kind of equation unless further modification is added to the scheme?

[TurbJet]

Originally Posted by FMDenaro

Well, I see that v0 produces a variation for x>=30 but the initial condition is upward. That makes me suppose that the starting of the compression region is downwind with respect to the used stencil [i-1,i]. Somehow, that does not respect the upwind criterion. I am not sure if this is the problem, just check the sign of the characteristic. This problem has differences from the standard non-linear Burgers equation.

But this interests me: for Lax-Friedrichs, it will never generate negative convective velocity. The convective velocity will always stop right before it reaches 0. I am wondering perhaps because the scheme includes stencils for both upwind and downwind...

[FMDenaro]

Originally Posted by TurbJet

But this interests me: for Lax-Friedrichs, it will never generate negative convective velocity. The convective velocity will always stop right before it reaches 0. I am wondering perhaps because the scheme includes stencils for both upwind and downwind...

The LF scheme is centred, therefore any direction is considered in the stencil.
The upwind scheme should be formulate in the general Godunov formulation, as illustrated in the book of Leveque. Maybe, you could also see the matlab sources of the clawpack software.
Have you tried to do the case with v0 is constant?

[TurbJet]

Originally Posted by FMDenaro

The LF scheme is centred, therefore any direction is considered in the stencil.
The upwind scheme should be formulate in the general Godunov formulation, as illustrated in the book of Leveque. Maybe, you could also see the matlab sources of the clawpack software.
Have you tried to do the case with v0 is constant?

Hm, I haven't tried v0 as constant. I'll give it a shot. I'll let you know.

[TurbJet]

Originally Posted by FMDenaro

The LF scheme is centred, therefore any direction is considered in the stencil.
The upwind scheme should be formulate in the general Godunov formulation, as illustrated in the book of Leveque. Maybe, you could also see the matlab sources of the clawpack software.
Have you tried to do the case with v0 is constant?

I tried constant v0, the upwind worked fine. But with constant v0, it just lost its original physical meaning. It just becomes simply transporting the wave.

[FMDenaro]

The first order upwind is strongly diffusive, try to work using a very fine mesh

[TurbJet]

Originally Posted by FMDenaro

The first order upwind is strongly diffusive, try to work using a very fine mesh

Actually what I was trying to say is that with constant v0, the wave does not have this sudden jump when reaches the small v0 region (congested area). I was not trying to say the diffusive property of upwind.

[FMDenaro]

Originally Posted by TurbJet

Actually what I was trying to say is that with constant v0, the wave does not have this sudden jump when reaches the small v0 region (congested area). I was not trying to say the diffusive property of upwind.

But for v0=const you have a different physics, there is no congestion. You see this example in a different section of the book of Leveque. Try to compare your solution with those in the book.

[TurbJet]

Originally Posted by FMDenaro

But for v0=const you have a different physics, there is no congestion. You see this example in a different section of the book of Leveque. Try to compare your solution with those in the book.

I mean no doubt that upwind will work for the case with v0=const. I was thinking of using upwind as low-resolution method for my flux-limiter for the original spatial-varying v0 case. Since upwind does not work for this, I guess I have to give it up and try other low-resolution scheme, may be Lax_Friedrichs

[FMDenaro]

Originally Posted by TurbJet

I mean no doubt that upwind will work for the case with v0=const. I was thinking of using upwind as low-resolution method for my flux-limiter for the original spatial-varying v0 case. Since upwind does not work for this, I guess I have to give it up and try other low-resolution scheme, may be Lax_Friedrichs

I think that upwind works also for variable v0, provided that is formulated in the correct way, that is in the Godunov form. Try to see the matlab source of the clawpack software