The Difference Equations

The difference equations are,

$$u_{n+\frac{1}{2}}^{(m+\frac{1}{2})} = u_{n+\frac{1}{2}}^{(m-\frac{1}{2})} + (P_{n}^{(m)} - P_{n+1}^{(m)})\delta t / \Delta m$$ (2.51)

$$x_{n+\frac{1}{2}}^{(m+1)} = x_{n+\frac{1}{2}}^{(m)} + u_{n+\frac{1}{2}}^{(m+\frac{1}{2})} \delta t \quad ,$$ (2.52)

where,

$$P_{n}^{(m)} = \mbox{constant} \times (\rho_{n}^{(m)})^\gamma \quad .$$ (2.53)

and,

$$\rho_{n}^{(m)} = \Delta m / {(x_{n+\frac{1}{2}}^{(m)} - x_{n-\frac{1}{2}}^{(m)})} \quad ,$$ (2.54)

$x_{n+\frac{1}{2}}^{(m)}$ and $u_{n+\frac{1}{2}}^{(m)}$ are the positions and velocities of the cell boundaries. $\rho_{n}^{(m)}$ and $P_{n}^{(m)}$ are the densities and pressures in the cells. $\Delta m$ is the mass in each cell. It is useful for the purpose of programming to redefine things to get rid of the various half integer indices. Hence we can write the equations as

$$u_{n}'^{(m+1)} = u_{n}'^{(m)} + (P_{n}^{(m)} - P_{n+1}^{(m)})\delta t / \Delta m$$ (2.55)

$$x_{n}'^{(m+1)} = x_{n}'^{(m)} + u_{n}'^{(m+1)} \delta t$$ (2.56)

$$\rho_{n}^{(m)} = \Delta m / {(x_{n}'^{(m)} - x_{n-1}'^{(m)})} \qquad ,$$ (2.57)

which maps more easily onto the arrays in a program.