Instabilities related to the geometric source term

jinwon park

For symmetric flows, I formulated symmetric 1-D Euler equations and axisymmetric 2-D Euler equations. To numerically solve the universal Euler equations with sources, I applied operator splitting method to the set of equations as dU/dt+dF/dr=S(r) in 1-D and dU/dt+dF/dr+dG/dz=S(r,z). For example, I first solve the standard Euler equation then the solution becomes the initial condition for source terms.

1. dUbar/dt+dF/dr=0 giving rise to Ubar

2. dU/dt=S

When I include the source terms for symmetric solutions, I suffered from stability troubles. However, when I do not include these, the code performed great. Could anyone advice me about the method I used?