Distributed surface roughness elements characterise Thermal Protection Systems (TPS) typical of supersonic and hypersonic flows. The presence of these distributed roughness elements cause an increase in drag and heat transfer.  As opposed to incompressible flow over roughness elements, there are very few experimental and numerical studies on supersonic flow over roughness. The most fundamental computational technique wherein, all scales of turbulence is resolved is Direct Numerical Simulation (DNS). The cost of performing DNS of fully resolved roughness is even higher than canonical DNS because of the refined mesh needed to solve the roughness elements.  To overcome this limitation, the current thesis aims at exploring low cost alternatives to DNS of fully resolved roughness for studying the effect of drag and heat transfer in supersonic flow over rough walls. DNS results from fully resolved full channel cube roughness for Mach 2 and Mach 4 at friction Reynolds number Re_τ = 500,1000  are analyzed. The results from the low cost models are compared against the fully resolved roughness simulated using full channels. Three lower-cost alternative, namely DNS of minimal channel flow of fully resolved roughness, DNS of modelled roughness and resolved RANS are considered. As for the DNS of minimal channel flow, it is found that the velocity shift ΔU⁺ is predicted accurately and therefore the added drag. However, it cannot be used to predict the temperature field because of lack of outer layer similarity for the thermodynamic statistics. As for the modeled roughness, an extension of the model by Busse and Sandham originally developed for incompressible flows is considered. In this case the roughness geometry is substituted by the additional drag and heat transfer that it induces on the flow, which take the form of source terms in the momentum and energy equations. We perform 17 DNS simulations with modeled roughness and compare the results to the fully resolved simulation. We find that the parametric forcing method is able to predict the velocity shift with good accuracy, although recovering the equivalent roughness height from the model parameters can only be done a posteriori. The model is able to qualitatively reproduce the temperature field, but thermodynamic statistics are inaccurate when compared to DNS of the fully resolved geometry. The final computational technique is RANS. In real case applications, RANS require the use of wall functions, and in the case of rough walls knowledge of the equivalent roughness height k_s⁺ is necessary. We attempt to see if RANS of fully resolved roughness can be used to estimate the velocity shift ΔU⁺ and therefore  k_s⁺ by limiting ourself to the  linear Spalart-Allmaras (SA) model. It is found to be inaccurate in computing the mean velocity profile at  k_s⁺≈ 40 with improvements in accuracy observed for k_s⁺≈ 80 when compared with the results from DNS for cube roughness element. However, the accuracy is still low to be used for estimating k_s⁺.