This chapter illustrates the behavior of high-order schemes using various test problems. Using potential enthalpy and entropy as the main thermodynamic variables has a number of advantages, especially in connection with strongly stratified flows near a central object with a deep potential well, which is relevant to studying outflow phenomena. The advantages of high-order finite difference scheme for astrophysical magnetohydrodynamic and turbulence simulations are highlighted. Particular emphasis is placed on the possibility of dynamo action in three-dimensional turbulent convection and shear flows, which is relevant to stars and astrophysical discs. A number of one-dimensional test cases are presented ranging from various shock tests to Parker-type wind solutions. Instead, in simple advection tests high-order methods are able to produce smoother solutions with less viscosity, which is important for accurate modeling of turbulence. The chapter describes some astrophysical applications of simulations using high-order schemes where hydromagnetic turbulence played an important role.