I am using the smoothed heaviside function to impose the circular I.C. on a cartesian mesh. There are a little stair-case effects in simulations. Could anyone advice me to prevent unphysical stair-case effects(zig-zag) in the case where a circular I.C. is embedded on a cartesian mesh?

If you are using a finite volume method, you should set the IC using cell averages of the initial condition. THat is the correct thing to do and will give you a better representation of the IC. FOr other schemes I dont know.

Could you tell me how to use cell averages of the initial condition? I am confusion of that term. Thanks in advance!

For any conserved quantity u(x,y,z) compute cell average for cell c as

U_c = Integral_over_c u(x,y,z) dx dy dz / volume(c)

and use this as initial condition in cell c. If initial condition is a simple function, then try to compute the integral analytically. Otherwise use some quadrature rule.

As you mentioned as "If initial condition is a simple function, then try to compute the integral analytically", the initial condition in my cases is simple. I do not understand how to compute the integral analytically. Could you teach me how to do? Thanks for your help.