How to avoid a non-physical effect in steady state

Alexander Starostin

My codes solve the problem of unplugging the volcanic conduit. It always has front with disappearin viscosity, under the front the viscosity grows exponentially and above the front the viscosity is 0. The flow stabilizes with non-zero discharge rate. The steady discharge profile along the conduit must be horizontal curve. But my numerical solution provides me a horizontal profile with a sag located in the front (In details, there is an elongated sag under the front and sligtly oscillating sharp peak above the front). I has applied several codes hoping to avoid the sag in the steady discharge: LxF, LxW, McCormick, Roe... But all of them produce the steady profile with sag.

Only the huge amount of cells in the mesh garantees the absence of the sag. But it leads to the days of calculations... The moving mesh may help but now I do not have enough time to build the code with such complicated method...

Maybe there is a better way out? How can I get my primitive horizontal line after the stabilization? Thank you before hand for having any ideas.