# Splitting Method

The non-homogenous 1D Euler equations with self-gravity are given by

which in short hand notation is

where

is vector of fluid variables, is their fluxes, and is the source-term vector.To solve this system of equations, we can employ operator splitting, which is analagous to the procedure for splitting the higher dimensional Euler equations (as I did for 2D — see wiki page).

It is basically as follows, first solve the homogenous equations,

with initial condition for the grid,

over a time-step dt (found by usual CFL condition for upwind Godunov scheme). This gives the solution,

Next, solve the equation for the source-term (again over time interval dt),

with initial condition

This gives the solution for the complete time-step,

Given the ODE's for the 'source' step only involve equations for momentum and energy, we conclude that only u and E change over this step (density does not).

Thus, a schematic is as follows,

**Data time level n ** **Hydro step**

**Source step**

# Update Formulas

To solve the first source term ODE,

replace with discrete derivatives,

which becomes,

since

we can just cancel the rho's and solve for the updated u,

This requires a dt (which is the same for the hydro step, solved using the CFL condition), and a gravitational potential phi. Phi is solved using the Jacobi solver, with the source function given by

and boundary conditions

To solve the 2nd source term ODE,

replace with discrete derivatives,

I am not exactly sure which time level to use for u (n+1 or n+½).

Rearranging for the updated version of the energy, we have,

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