This work presents data assimilation (DA) strategies implemented to correct compressible RANS models in a high-order discontinuous-Galerkin (DG) framework. We present four field-inversion formulations to treat highly compressible flows: the first one is based on source terms in the momentum and energy equations; the second one is an eddy-viscosity a posteriori correction, where we aim at fixing the eddy-viscosity alone; the third and fourth ones consist respectively in a source term in the Spalart-Allmaras transport equation and a tune of the turbulence production. A constrained optimisation problem is defined to find the optimal degrees of freedom of these parameters in the DG function space. The discrete adjoint approach, consistent with the formal problem, is used to compute cost function gradients. The DA approaches are tested on a shock-wave/turbulent boundary-layer interaction configuration. A significant correction of the mean flow is achieved when source terms are introduced in the momentum and energy equations. Moreover, wall variables such as the skin-friction and pressure coefficient are fully reconstructed if the entire DNS velocity field is taken into account during the assimilation. Further investigations concerning the optimizer are made by comparing quasi-Newton optimizers, as the L-BFGS algorithm, and stochastic optimization methods, as the Adam algorithm. The discrepancy between the two optimizers shows existence of local minima. For this reason, a regularization of the DA parameter space is proposed with penalization techniques.