  When fully conservative methods are used to simulate transcritical flow,
spurious pressure oscillations and numerical instability are generated. The
strength and speed of propagation of shock waves cannot be represented
correctly using a semi-conservative or primitive method. In this research, an
adaptive primitive-conservative scheme is designed to overcome the aforesaid
two difficulties. The underlying cause for pressure oscillation is analyzed
within the framework of Finite Volume Method (FVM). We found that the
nonlinearity of the thermodynamic properties of transcritical fluids renders
standard conservative numerical methods ineffective. In smooth regions, schemes
based on primitive variable are used to eliminate spurious pressure
oscillations. For the purpose of correctly capturing shock waves, the modified
Roe Riemann solver for real fluid is utilized in regions where shock waves
induce discontinuity. The adaptive numerical approach relies only on the speed
of sound, eliminating the requirement to calculate the derivatives of
thermodynamic quantities. A large number of numerical test cases conducted in
one- and two-dimensional spaces have shown the robustness and accuracy of the
proposed adaptive scheme for the simulations of high speed transcritical flows.
