We present direct numerical simulations of the shock wave boundary layer interaction (SBLI) at Mach number 2.9 over a 24° ramp. We study both the numerical accuracy and flow physics. Two classes of spatial reconstruction schemes are employed: the monotonic upstream-centered scheme for conservation laws and the Weighted Essentially Non-Oscillatory (WENO) scheme, of accuracy ranging from 2nd- to 11th-order. Using the canonical Taylor-Green vortex test-case, a simple and computationally inexpensive rescaling of the candidate stencil values - within the context of the high-order WENO scheme - is proposed for reducing the numerical dissipation, particularly in under-resolved simulations. For the compression ramp case, higher-order schemes are shown to capture the size of the SBLI separation zone more accurately, a consequence of resolving much finer turbulence structures. For second- and fifth-order schemes, the energy of the unresolved small scale turbulence shifts the cascade of the turbulence kinetic energy (TKE) spectrum, thus resulting in more energetic large scale turbulent structures. Consequently, the λ-shock foot shifts further downstream, leading to a smaller separation bubble size. Nonetheless, other statistical quantities, such as the turbulence anisotropy invariant map and the turbulence kinetic energy budget terms, show little dependence on the type and order of the spatial reconstruction scheme. Finally, using the more accurate ninth-order WENO results, it is reasoned that the interaction of the λ-shock with the post-shock relaxation region drives the low-frequency oscillation of the λ-shock.