Instabilities related to the geometric source term
 

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?

Re: Instabilities related to the geometric source
 

Why dont you directly solve including the source term ? Is the source term an implicit function of the primitive variables? or just some terms of the form rho*v_r etc?

What time scheme do you use? A Runge-Kutta or a simple leap-frog? or an implicit method?

I am not sure I understand the reason behind performind such a splitting ?

Re: Instabilities related to the geometric source
 

The source terms are jsut some terms such as rho*v as you told. The reason I solved it by splitting methods was based on the statement in the literature. In that literature(E.F. Toro, Riemann problem...), the operator splitting method is only one suggested. Can we solve it by direct coupling with advection terms? If so, it will be better to the simple implementation. Have you seen any literature recommending for coupled solution approach?

Re: Instabilities related to the geometric source
 

Hi, sorry for being late to respond...

Are solving the equations with an implicit time scheme or an explicit ?

If it is explicit then this is straightforward, if implicit then it involves something like inversing a matrix (or some iteration of some kind).

I have use spliting only to split space dimensions and to split "implicit" terms from "explicit" terms (again as reltated to time scheme).

I must confess that it is kind of difficult to follow the problem without seeing the original equations and the numerical equations.

Re: Instabilities related to the geometric source
 

I am using an explicit scheme. Based on the previous post, I substitute the source term into the right hand side. I am testing many cases. In 1-D, it worked well. Thanks for your reply.

Jinwon