Numerical analysis of time accuracy of a primitive variable-based formulation of the conservative form of the governing equations for compressible flows

A class of Computational Fluid Dynamics codes uses primitive variable-based formulation to solve the conservative form of the governing equations. Under certain specific conditions, the approach yields inaccurate results for unsteady problems. Here, Euler equations are used to numerically analyse time accuracy when using such an approach. It is demonstrated, using a simple shock tube problem, that the primitive variable-based approach does not strictly preserve time accuracy. A dual-time stepping technique, which uses an inner iteration of pseudo-time within the physical time, is used to correct the issue. The method is applied to the shock tube problem, which eliminates the time-dependent discrepancies. The problem, and the correction, is also applied to a supersonic flow over a wedge. In both cases, the results are shown to be identical to the solution obtained using a conservative variable-based approach.