Application of the Method of Fractional Steps to Hyperbolic Equations

Abstract

Consider the equation of acoustics

$$\frac{{\partial u}}{{\partial t}} - {a^2}\frac{{\partial v}}{{\partial x}} = 0;\,\frac{{\partial v}}{{\partial t}} - \frac{{\partial v}}{{\partial t}} = 0,$$

(3.1.1)

where u is the velocity; v is the specific volume; a is the velocity of sound, and x; is the Lagrangian coordinate. Written in terms of Riemann invariants

$$r = u - av;\,s = u + av,$$

(3.1.2)

Eq. (3.1.1) takes the form

$$\frac{{\partial r}}{{\partial t}} + a\frac{{\partial r}}{{\partial x}} = 0;\,\frac{{\partial s}}{{\partial t}} - a\frac{{\partial s}}{{\partial x}} = 0$$

(3.1.3)