Conservation Laws

Abstract

The class of conservation laws is a very important class of partial differential equations because as their name indicates, they include those equations that model conservation laws of physics (mass, momentum, energy, etc.). In Sections 1.6, 4.2.2 and 5.8.1, we used “the conservation law approach” to derive difference equation approximations to certain linear partial differential equations. This approach was related to the subject of this chapter in that we considered the equation as if it had come from some conservation law and proceeded to derive a difference approximation that would respect the conservation principle. The added difficulty that we shall address in this chapter is that conservation laws are generally nonlinear. As we shall see, this strongly affects both the solution’s behavior and the numerical solution.