I am solving the cylindrical wave propagation problem on the cartesian mesh.

Since the mesh is in the cartesian, the defining of cylindrical I.C. is was trivial.

I used the smoothed heaviside function as follows

FUNCTION heaviside(x,dx,dy,eps) IMPLICIT DOUBLE PRECISION(a-h,o-z) PARAMETER(PI=3.141598d0) IF(x .LT. -eps) THEN

heaviside=0.d0 ELSEIF(x .GE. eps) THEN

heaviside=1.d0 ELSE

heaviside=0.5d0*(1.d0+x/eps+1.d0/PI*sin(PI*x/eps)) ENDIF ENDFUNCTION heaviside

It worked well when the discontinuity is not serious(e.g. 1 to 0.1 or 2 to 0.2). However, when I solved a large gradient of a variable(e.g. 500 to 0.1 or 10e9 to 10e4), the initial condition became strange.

Even though the smoothed heaviside function gave a smoothed variation like 0.0,0.0001,0.85,1.000, a strange set of I.Cs were created.

That is, due to large gradient, a strange sharp hump was observed in the initial condition.

I wonder if anyone advice me how to define a circular I.C on a cartesian mesh.

Please help me... I need your help. Really.

(probably you like to have sharp interface)

you can define your cylinderical wave by zero level set of an scaler function (so you sholud define a distance field for initial configuration) and then use either diffusion-less HJ-solvers or use immersed boundary methods (i think S. H. Udaykumar work JCP 05 is suitable for you) for monitoring its propagation.

>Since the mesh is in the cartesian, the defining of cylindrical I.C. is was trivial

?? Do u understand what u say? (poor english !)

The famous solution is the level set immersed boundary, i agree with the previous thread, the mentioned paper (there are seris of 3-papers, second one: http://dx.doi.org/10.1016/j.jcp.2005.03.032) is a good ref. complete details is presented (note that this Ref. is very famous and was in 25-top paper list of JCP).