For certain problems, especially when used to analyze compressible flow in a duct or in case the flow is cylindrically or spherically symmetric, the one-dimensional Euler equations are a useful first approximation. Generally, the Euler equations are solved by Riemann 's method of characteristics. This involves finding curves in plane of independent variables (i.e., x and t ) along which partial differential equations (PDEs) degenerate into ordinary differential equations (ODEs). Numerical solutions of the Euler equations rely heavily on the method of characteristics.