Compressible inviscid 1d spherical symmetry flow
 

I am solving 1d Euler spherical symmetry flow.

U_t+F_r=-alpha/r*F(U)

in other forms, rho_t+(rho u)_r=-alpha/r*(rho u) rhou_t+(rhou^2+p)_r=-alpha/r*(rho u^2) rhoe_t+(u(rhoe+p))_r=-alpha/r*(u(rhoe+p))

To solve this inhomogeneous hyperbolic equation, I used the fractional-step method for two equations

U_t+F_r=0 ! homogeneous conservation laws U_t=-alpha/r*F(U) ! ODE for the geometric source term

3rd RK and 2nd RK are employed for the first and the second equation above. I am thinking r is the distance from the orgin. That is, r is the same as the x in the cartesian system if the origin is located at the left-end.

Unfortunately, I couldn't obtain a desirable results without the oscillation.

I was trying to find any book containing this kind of problem but I couldn't find it.

Is there any misunderstanding about aboves? Have you seen any good reference how to discretize the geometric source term? Currently, I am using DGM method.

Please advice me. Thanks in advance!