Time accuracy analysis of primitive variable-based conservative form governing equations

The governing equations using primitive properties of pressure, velocity and temperature as dependent variables, but constructed in the conservative form can be applied for calculating solutions for steady state problems. When the method was recently used to simulate a canonical shock-tube problem, however, overshoot of temperature was observed after the shock. Moreover, the errors cannot be eliminated by using fine grid, high spatial order of accuracy or any alternative inviscid fluxes schemes that are available, implying the numerical discrepancy may be caused by the method itself. Numerical analysis was conducted on the method using the one-dimensional Euler equations as the model equation system. It can be shown that the numerical error specifically arises from the discretized time terms. A dual-time-looping technique was developed to address the issue. It used conservative variables in physical-time derivatives while primitive variables for pseudo-time terms. An inner iteration procedure within two adjacent physical-time steps were driven until a steady state was reached. The resulting governing equation converged to the corresponding conservative form in time, and the time-accurate solution was recovered. © 2019, NUDT Press. All right reserved.