[yidongxia]

A simple 1D scalar linear advection can be described as below:

du/dt + a*du/dx = 0

where u = u(x,t) is the variable concerned, and a is the propagation velocity. Suppose a step function profile is used as initial condition in x = [0, 1], where u(x,0) = 1 for x=[0, 0.1], and u(x,0) = 0 for x=[0.1, 1].

First, I am aware that methods like finite volume with upwind scheme and flux/slope limiting can do pretty well. However, I need to solve this problem within the scope of continuous FEM. Then if a regular SUPG (streamline upwind Petrov Galerkin) formulation is applied, I can get stabilized solution, but still have overshoots and undershoots near the vicinity of the discontinuity, which is commonly known. I tried a few approaches from papers in the 80's to eliminate the shoots, but the results are not satisfying.

Any recommended literature for a direct guidance on this problem, or a quick write-up of formula for adding some discontinuity capturing based artificial diffusivity to achieve a shoots-free solution?

Thanks for any comment!

[yidongxia]

___
----|
----|______________________

Initial profile is like above.

[Rami]

Many years ago I used an artificial viscosity scheme to smear shocks based on S.W. Key's in my papers on dynamic shell FEM. Look for these papers in Computers & Structures (1990?), and mainly Key's reference therein.