Introduction

This chapter concerns simple scalar models of the Euler equations, called scalar conservation laws. Scalar conservation laws mimic the Euler equations, to the extent that any single equation can mimic a system of equations. In order to stress the parallels between scalar conservation laws and the Euler equations, the first part of this chapter essentially repeats the last two chapters, albeit in a highly abbreviated and simplified fashion. As a result, besides its inherent usefulness, the first part of this chapter also serves as a nice review and reinforcement of the last two chapters.

Like the Euler equations, scalar conservation laws can be written in integral or differential forms. In integral forms, scalar conservation laws look exactly like Equation (2.21) except that the vector of conserved quantities u is replaced by a single scalar conserved quantity u, and the flux vector f(u) is replaced by a single scalar flux function f(u). Similarly, in differential forms, scalar conservation laws look exactly like Equations (2.27) and (2.30) except that the vector of conserved quantities u is replaced by a single scalar conserved quantity u, the flux vector f(u) is replaced by a single scalar flux function f(u), and the Jacobian matrix A(u) is replaced by a single scalar wave speed a(u), not to be confused with the speed of sound.

Replacing several interlinked conserved quantities by a single conserved quantity dramatically simplifies the governing equations while retaining much of the essential physics. In particular, scalar conservation laws can model simple compression waves, simple expansion waves, shock waves, and contact discontinuities. Like simple waves in the Euler equations, scalar conservation laws have a complete analytical characteristic solution.