Conservation Laws

Publisher Summary This chapter discusses conservation laws applicable to partial differential equations. These laws yield quantities that remain invariant during the evolution of the partial differential equation. Given an evolution equation, which is a partial differential equation of the form ut = F(u, ux, uxx,….) a conservation law is a partial differential equation of the form ∂T/∂t (u(x, t) + ∂X/∂x (u(x, t) = 0 which is satisfied by all solutions of ut = F(u, ux, uxx,….). Conservation laws allow estimates of the accuracy of a numerical solution scheme. Not all partial differential equations have an infinite number of conservation laws, there may be none or a finite number. A conservation law for an evolution equation is called trivial if T is itself, the x derivative of some expression. If the equation ut = F(u, ux, uxx,….) has an infinite sequence of nontrivial conservation laws, then the equation is formally integral. If a given partial differential equation is not written in conservation form, there are a number of ways of attempting to put it in a conserved form.