In the shock-capturing scheme of Ref. [10], the inviscid part of the numerical flux is computed as a convex combination of three types of fluxes, *F*^{A}, *F*^{D} and *F*^{C}. The *F*^{A} is dissipative and *F*^{D} and *F*^{C} are less dissipative; *F*^{D} can capture a stationary contact discontinuity without inner points. The arguments of *F*^{A} and *F*^{D} are computed by MUSCL with the van-Leer slope limiter and will be hereafter denoted by \( {\boldsymbol{h}}_{MUSCL}^L \) and \( {\boldsymbol{h}}_{MUSCL}^R \), where *h* stands for the primitive variables, i.e. *h =**ρ*, *u*, *v*, *w*, *P* (density, flow velocity, and pressure). The arguments of *F*^{C} are computed from fifth order accurate Lagrange’s polynomial approximations of primitive functions for conservative variables; the scheme is developed for structured grids. The *F*^{A} is for shock capturing, *F*^{D} is for contact-discontinuity capturing and *F*^{C} is for smooth regions. The reader is referred to Ref. [10] for the definitions of these numerical fluxes. The weights in the combination are controlled accordingly. The numerical flux for the Navier-Stokes equations is simply made by the addition of the physical dissipation, which is approximated by a conventional second-order accurate central finite difference formula.

In order to reduce computational cost, we introduce several simplifications into the original high-order accurate shock-capturing scheme. The reconstruction is based on cell-averaged values in the simplified scheme. The number of Gauss points needed for the integration of the numerical flux over cell-face is reduced (from four in 3D case) to one and the formal accuracy of the time marching is reduced to second order (RK-2). The inviscid numerical flux is computed as a convex combination of *F*^{A} and *F*^{D}.

$$ \boldsymbol{F}=\alpha {\boldsymbol{F}}^A+\left(1-\alpha \right){\boldsymbol{F}}^D, $$

where *α* is a weight in the range of 0 ≤ *α* ≤ 1. We prepare two schemes. The first scheme, (S1), the arguments of *F*^{A} and *F*^{D} are common and are \( {\boldsymbol{h}}_{MUSCL}^L \) and \( {\boldsymbol{h}}_{MUSCL}^R \). The second scheme, (S2), also uses \( {\boldsymbol{h}}_{MUSCL}^L \) and \( {\boldsymbol{h}}_{MUSCL}^R \) as the arguments of *F*^{A} but those of *F*^{D} are modified. Let the arguments of *F*^{D} in (S2) be denoted by \( {\boldsymbol{h}}_{S2}^L \) and \( {\boldsymbol{h}}_{S2}^R \) and let *h*_{k} be the value of *h* at the cell-centre of the index *k*, which is computed from the cell-average of the conservative variables. A quartic function generated from *h*_{k} (*k* = *i* − 2, *i* − 1, *i*, *i* + 1, *i* + 2) provides \( {\boldsymbol{h}}_{Quartic}^R \) at the cell-face *i* − 1/2 and \( {\boldsymbol{h}}_{Quartic}^L \) at the cell-face *i* + 1/2. The \( {\boldsymbol{h}}_{S2}^H\ \left(H=L,R\right) \) at each cell-face is computed as a convex combination of \( {\boldsymbol{h}}_{MUSCL}^H \) and \( {\boldsymbol{h}}_{Quartic}^H \) there:

$$ {\boldsymbol{h}}_{S2}^H=\beta {\boldsymbol{h}}_{Quartic}^H+\left(1-\beta \right){\boldsymbol{h}}_{MUSCL}^H, $$

where *β* is a weight in the range of 0 ≤ *β* ≤ 1. The weights *α* and *β* are computed at each cell-face according to the physical situation and the smoothness of solution. The guiding principle is as follows: (i) *F*^{A} should be dominant at cell-faces around shock waves and *F*^{D} should be so except around them; (ii) \( {\boldsymbol{h}}_{MUSCL}^H \) should be dominant in \( {\boldsymbol{h}}_{S2}^H \) at cell-faces in regions where *h* changes abruptly in the scale of cell width, e.g. contact discontinuities, and \( {\boldsymbol{h}}_{Quartic}^H \) should be so in regions where *h* changes slowly in the scale of cell width. In the present study the following formulas for *α* and *β* are adopted.

$$ \alpha =1-\exp \left[-{C}_1\frac{\mid {P}^L-{P}^R\mid }{P^L+{P}^R}\right], $$

$$ \beta =\exp \left[-\left({C}_2\frac{\mid {P}^L-{P}^R\mid }{P^L+{P}^R}+{C}_3\frac{\mid {\rho}^L-{\rho}^R\mid }{\rho^L+{\rho}^R}+{C}_4\frac{\mid {u}^L-{u}^R\mid }{\sqrt{T_{\ast }}}+{C}_5\frac{\mid {v}^L-{v}^R\mid }{\sqrt{T_{\ast }}}+{C}_6\frac{\mid {w}^L-{w}^R\mid }{\sqrt{T_{\ast }}}\right)\right], $$

where *h*^{H} (*h =**ρ*, *u*, *v*, *w*, *P*, *H* = *L*, *R*) are computed by MUSCL (the subscripts MUSCL is omitted for concise expression), \( {T}_{\ast }=\frac{T^L+{T}^R}{2} \) (*T* is the temperature, computed from *ρ* and *P* by EOS, e.g. *T* = *P*/*ρ*), and *C*_{k} (*k* = 1, ⋯, 6) are positive constants [*C*_{k} = 10 (*k* = 1, ⋯, 6) in the present study]. Since the weight *β* is determined from the magnitude of the jump \( \mid {\boldsymbol{h}}_{MUSCL}^L-{\boldsymbol{h}}_{MUSCL}^R\mid \), the numerical dissipation of *F*^{D} could be insufficient for the elimination of waves with high wave numbers such as those with zigzag wave forms when their amplitudes are very small; the jump \( \left|{\boldsymbol{h}}_{S2}^L-{\boldsymbol{h}}_{S2}^R\right| \) could be much smaller than their amplitudes for such waves. This is in contrast to the original scheme.