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, FA, FD and FC. The FA is dissipative and FD and FC are less dissipative; FD can capture a stationary contact discontinuity without inner points. The arguments of FA and FD 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 FC are computed from fifth order accurate Lagrange’s polynomial approximations of primitive functions for conservative variables; the scheme is developed for structured grids. The FA is for shock capturing, FD is for contact-discontinuity capturing and FC 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 FA and FD.
$$ \boldsymbol{F}=\alpha {\boldsymbol{F}}^A+\left(1-\alpha \right){\boldsymbol{F}}^D, $$
where α is a weight in the range of 0 ≤ α ≤ 1. We prepare two schemes. The first scheme, (S1), the arguments of FA and FD 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 FA but those of FD are modified. Let the arguments of FD in (S2) be denoted by \( {\boldsymbol{h}}_{S2}^L \) and \( {\boldsymbol{h}}_{S2}^R \) and let hk 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 hk (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) FA should be dominant at cell-faces around shock waves and FD 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 hH (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 Ck (k = 1, ⋯, 6) are positive constants [Ck = 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 FD 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.