The behaviour of an ideal compressible gas is described by the *Euler equations*:

(1) | |||

(2) | |||

(3) |

with

(4) |

Reducing these to one spatial dimension and using cartesian
coordinates one can write them as

(5) |

The Euler equations are not complete without an equation of state. We
choose an ideal gas for which

(6) |

(7) |

A more general form of the Euler equations is

(8) |

We will be using the spherical coordinate . For this the momentum
equation is

(9) |

(10) | |||

(11) | |||

(12) |

is still the vector of conserved quantities vector since it is total mass, momentum and energy which are conserved, not their densities. In spherical coordinates a volume element is proportional to (really but we are not considering the coordinate here), so by multiplying the densities by the position dependent volume factor , one obtains the conserved quantities.

One sees that choosing a non-cartesian coordinate leads to the appearance of a geometrical source term. There are actually several ways in which to split off the geometrical source terms, but here we choose the one shown in Eqs. (10)-(12).