# Chapter 18

# The Numerical Method of Characteristics

## 18.1. Introductory remarks

The integration technique of Euler equations by the method of characteristics is based on the use of equations [17.13] of the previous chapter that have been linearized and then discretized. Different operators are used depending on whether the point is calculated within the fluid or located on a boundary (wall, axis of symmetry or centerline, and fluid boundary). The procedure is similar in spirit to that used for one-dimensional unsteady flows (see section 24.1). As in any numerical method, the solution to the equations is sought in discrete points, which together form a mesh, chosen close enough to each other so that the flow properties vary only slightly when we pass from one point to one of its neighbors. This condition justifies the linearization of the equations and ensures the convergence of the calculation under certain conditions that we will not discuss here.

The method of characteristics is to calculate the flow step-by-step, progressing along the characteristic lines (*η*) and (*ξ*). In this integration technique, the meshing depends on the solution itself since (*η*) and (*ξ*) are functions of local properties of the flow. Thus, this meshing is constructed as the calculation progresses, which introduces some algorithmic complexity into the structure of a computer program, especially when shock waves are formed within the flow. However, the method ensures correct transmission of information in agreement ...