# Chapter 24

# Numerical Methods for Calculating Unsteady Flows

## 24.1. Integration by the method of characteristics

There are circumstances where the solution to the problem cannot be obtained as simply as in the cases that have been analyzed, for example, if the boundary conditions vary with time, or if the piston moves at a variable speed. It is then necessary to resort to numerical integration techniques by determining the solution at discrete points of the space [*x*, *t*] and proceeding step by step in the calculation domain. Under these conditions, the solution is obtained only at the calculation points and has an approximated character, since the equations are not solved exactly. This is a shortcoming of all numerical methods, widely used today, but it is possible to get a high enough accuracy by refining the calculation grid; i.e. by considering calculation points very close to each other and using accurate discretization schemes. The method of characteristics, described at length in Chapter 18 dedicated to steady supersonic flows, builds on the mathematical properties of unsteady flows which we have just seen.

We assume that in the plane [*x*, *t*], the flow is, as shown in Figure 24.1a, known at two points (1) and (2). The intersection of the characteristic (*η*) through (1) and of the characteristic (*ξ*) through (2) is a point (3) whose properties are related to those in (1) and (2) by the Riemann relations:

These two relations allow the calculation of *u*_{3} and *a*_{3} through:

The position ...