problem simulating an advection equation

hami9293 · July 17, 2015, 16:51

Hello,

I am struggling with a very simple problem for the past few days. I won't be surprised if the answer is terribly simple.

I want to simulate a simple advection equation:

du/dt + U du/dx =0
u(t=0) = sin(x)
u(x=0, x = pi) =0

I wrote a simple finite difference upwind spatial discretization with explicit time advance
$u^{n+1}_{j}$ = $u^{n}_{j}$ - $\Delta$ t .U/ $\Delta$ x ( $u^{n}_{j}$ - $u^{n}_{j-1}$ )

The problem is that the amplitude of the waves shrinks as time goes by, while I expect the wave shape is preserved and only translated to the left (assuming U>0). (see figure 1)

https://www.dropbox.com/s/nrvadimqenf2uxx/1.bmp?dl=0

Alternatively, I can use a more sophisticated scheme such as beam_warming.
The shrinking problem here is fixed, but I get some wobbling noises.
see figure

https://www.dropbox.com/s/ric984qv73xvh9w/2.bmp?dl=0
Any idea/explanation/suggestion/comment would be great.

Thanks for reading.

FMDenaro · July 17, 2015, 17:01

what you discovered is the well known presence of artificial viscosity in the first order upwind method....I suggest a carefull reading of a good text of CFD to study the theoretical reasons by means of the modified equation.

higher order schemes are not monotone and can generate wiggles

July 17, 2015, 16:51	problem simulating an advection equation	#1
hami9293 New Member Terry Tensinsky Join Date: Feb 2010 Posts: 14 Rep Power: 16	Hello, I am struggling with a very simple problem for the past few days. I won't be surprised if the answer is terribly simple. I want to simulate a simple advection equation: du/dt + U du/dx =0 u(t=0) = sin(x) u(x=0, x = pi) =0 I wrote a simple finite difference upwind spatial discretization with explicit time advance $u^{n+1}_{j}$ = $u^{n}_{j}$ - $\Delta$ t .U/ $\Delta$ x ( $u^{n}_{j}$ - $u^{n}_{j-1}$ ) The problem is that the amplitude of the waves shrinks as time goes by, while I expect the wave shape is preserved and only translated to the left (assuming U>0). (see figure 1) https://www.dropbox.com/s/nrvadimqenf2uxx/1.bmp?dl=0 Alternatively, I can use a more sophisticated scheme such as beam_warming. The shrinking problem here is fixed, but I get some wobbling noises. see figure https://www.dropbox.com/s/ric984qv73xvh9w/2.bmp?dl=0 Any idea/explanation/suggestion/comment would be great. Thanks for reading.

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July 17, 2015, 17:01		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,897 Rep Power: 73	what you discovered is the well known presence of artificial viscosity in the first order upwind method....I suggest a carefull reading of a good text of CFD to study the theoretical reasons by means of the modified equation. higher order schemes are not monotone and can generate wiggles