Appendix: Basics of Godunov Methods

A system of equations n conservation form generically give rise to shocks (discontinuities on the variables describing the state of the fluid) even starting from smooth initial data. This implies that discretizations based on the continuity of the functions, like finite difference methods, will fail. There are different approaches we could take to solve this system. Here we will take a finite volume approach, we will assume that we have a mesh of grid points that define a cell structure of our space-time (see Figure 10).

Figure 10: Grid Discretization

In the presence of discontinuities the only way to make sense of our system of equations is to consider the average over a finite volume of the space-time. Therefore to find the discretization we take the average of equation (32) over a grid cell ${\mathcal C}^{n+1/2}_i$.

$$\frac{1}{V_{{\mathcal C}^{n+1/2}_i}}\int_{{\mathcal C}^{n+1/... ...1/2}_i}}\int_{{\mathcal C}^{n+1/2}_i} {\mbox{\boldmath$\psi$}}$$ (66)

where ${\mathcal C}^{n+1/2}_i$ is the region of space-time defined by ${(t^n,t^{n+1/2})\cap(x_{i-1/2},x_{i+1/2})}$, and $V_{{\mathcal C}^{n+1/2}_i} = \Delta t \Delta x$ is its volume. The resulting equation can be written as:

$$\frac{1}{\Delta x \Delta t} \int_{x_{i-1/2}}^{x_{i+1/2}} \i... ..._{i+1/2}} \int_{t^n}^{t^{n+1}} {\mbox{\boldmath$\psi$}} dx dt.$$ (67)

We can integrate the different terms of the equation using Gauss theorem:

$$\frac{\bar {\bf q}_i^{n+1} - \bar {\bf q}_i^{n}}{\Delta t} +... ...\Delta x} = {\widetilde {\mbox{\boldmath$\psi$}}}^{n+1/2}_{i}.$$ (68)

Here we have used the following definitions: the spatial averages of the conservation variables,

$$\bar {\bf q}^n_i \equiv \frac{1}{\Delta x} \int_{x_{i-1/2}}^{x_{i+1/2}} {\bf q}(t^n,x) dx,$$ (69)

the temporal average for the fluxes, also referred as the numerical fluxes,

$${\bf F}^{n+1/2}_{i+1/2} \equiv \frac{1}{\Delta t} \int_{t^n}^{t^{n+1}} {\bf f}\left({\bf q}(t,x_{i+1/2})\right) dt,$$ (70)

and the total average over the space-time cell for the sources,

$${\widetilde {\mbox{\boldmath$\psi$}}}^{n+1/2}_{i} \equiv \fr... .../2}} \int_{t^n}^{t^{n+1}} {\mbox{\boldmath$\psi$}}(t,x) dx dt.$$ (71)

The idea now is to use equation (70) to calculate $\{\bar{\bf q}^{n+1}_i\}$ assuming we know the values $\{\bar{\bf q}^n_i\}$. However the calculation of the numerical fluxes $\{{\bf F}^{n+1/2}_{i+1/2} \}$ is not as straight forward as one may think, these fluxes are averages on time so in order to explicitly calculate them we need to know the solution before hand. In addition to that, and more importantly, the values for the fluid quantities on the left side of the cell boundary $\{\bar{\bf q}^n_i\}$ and on the right side $\{\bar{\bf q}^n_{i+1}\}$ won't agree in general. The values have discontinuities and a priori is not clear which values to use in order to compute the numerical fluxes. One way to solve these problems is to use an idea due to Godunov which implies solving a Riemann problem at every cell boundary in order to calculate $\{{\bf F}^{n+1/2}_{i+1/2} \}$. For more information about Godunov methods see [5]. In the following section we explain one of this methods.