High-order accuracy: Monotonicity and non-linear methods

These lecture notes are exclusively concerned with numerical methods for scalar homogeneous hyperbolic equations in one space dimension. We study monotonicity and some other basic properties of numerical methods. We state and prove Godunov's theorem and then go on to construct two classes of non-linear methods to circumvent it, TVD and ENO methods. Two examples of non-linear methods are given: the MUSCL-Hancock method, second order, and the ADER method, of arbitrary order.