Finite-difference methods for the advection equation

A conservation law is an expression in mathematical terms of the balance within a physical system. It is a statement that the production of a physical quantity such as mass, energy or charge in a closed volume is exactly equal to the flux of that quantity across the boundary of that volume. Such conservation laws often take the form of partial differential equations with appropriate boundary conditions or equivalent integral forms. The IBVP is ill-posed if a > 0 and we set the boundary Dirichlet boundary condition U(L, t) = g(t) where g is a continuous time-dependent function. To compute the solution of (2) we can of course also use other techniques such as Fourier series and Laplace transforms.