  In this chapter, we aim at presenting the basic techniques necessary to go
beyond the widely accepted paradigm of second-order numerics. We specifically
focus on finite-volume schemes for hyperbolic conservation laws occuring in
fluid approximations such as the equations of ideal magnetohydrodynamics or the
Euler equations of gas dynamics. For the sake of clarity, a simple fourth-order
ideal magnetohydrodynamic (MHD) solver which allows to simulate strongly
shocked systems serves as an instructive example. Issues that only or mainly
arise in the world of higher-order numerics are given specific focus.
Alternative algorithms as well as refinements and improvements are dicussed and
are referenced to in the literature. As an example of application, some results
on decaying compressible turbulence are presented.
