2.5 Introduction to Finite Volume Methods

2.5.5 Finite Volume Method for Nonlinear Systems

Measurable Outcome 2.1, Measurable Outcome 2.3, Measurable Outcome 2.4

The basic finite volume approach can be extended to nonlinear systems of equations such as the Euler equations. The main issue in this extension is how to calculate an upwind flux when there is a system of equations. In one dimension, the basic finite volume discretization remains the same as given by Equation 2.108,

\[\Delta x_ i \frac{U_ i^{n+1} - U_ i^ n}{{\Delta t}} + F_{i+\frac{1}{2}}^{n} - F_{i-\frac{1}{2}}^ n = 0.\] (2.117)

The flux, however, must upwind (to some degree) all of the states in the equation. One relatively simple way in which this can be done is using what is known as the local Lax-Friedrichs flux. In this case, the flux is given by,

\[F_{i+\frac{1}{2}}(U_ i, U_{i+1}) = \frac{1}{2}\left[F(U_{i+1}) + F(U_{i})\right] - \frac{1}{2}s_{\max }\left(U_{i+1} - U_{i}\right), \label{equ:upwindflux_ laxfriedrichs}\] (2.118)

where \(s_{\max }\) is the maximum speed of propagation of any small disturbance for either state \(U_ i\) or \(U_{i+1}\).

Lax-Friedrichs Flux for 1-D Euler Equations

For the one-dimensional Euler equations, there are three equations which are approximated, i.e. conservation of mass, conservation of \(x\)-momentum, and conservation of energy. A small perturbation analysis can be performed which shows that the three speeds of propagation for this set of equations are \(u\), \(u-a\), and \(u+a\) where \(u\) is the flow velocity and \(a\) is the speed of sound. Thus, the maximum speed will always be \(|u| + a\) and the corresponding value of \(s_{\max }\) for the Lax-Friedrichs flux is,

\[s_{\max } = \max \left(|u|_{i} + a_{i}, \, |u|_{i+1} + a_{i+1}\right).\] (2.119)