Modified Ghost Fluid Method with Acceleration Correction (MGFM/AC)

Abstract

In this work, we show that the modified ghost fluid method might suffer overheating and leads to inaccurate numerical results when directly applied to a moving rigid boundary with acceleration. We discover the insightful reasons and then develop a new technique to take into account the effect of boundary acceleration on the definition of ghost fluid states based on a generalized Piston–Riemann problem. Theoretical analysis and numerical results show that the modified ghost fluid method with acceleration correction can overcome such difficulty effectively.

Appendices

Appendix A: Basic Differential Relations and the Rankine–Hugoniot Relations for Fluid Flow

To solve the Riemann problem and the generalized Riemann problem, we need to use the concept of Riemann invariants and provide some differential relations among different variables. For more details, we can refer to [35, 39, 40].

For this purpose, we write the system of Euler equations (2.1) in the following form:

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {\frac{{\mathrm{{D}}\rho }}{{\mathrm{{D}}t}} = - \rho \frac{{\partial u}}{{\partial x}}}\\ {\frac{{\mathrm{{D}}u}}{{\mathrm{{D}}t}} = - \frac{1}{\rho }\frac{{\partial P}}{{\partial x}}}\\ {\frac{{\mathrm{{D}}S}}{{\mathrm{{D}}t}} = 0}, \end{array}} \right. \end{aligned}$$

(A.1)

where $D/Dt=\partial /\partial t+u\cdot \partial /\partial x$ is the material derivative, and the entropy S is related to the other variables through the second law of thermodynamics

$$\begin{aligned} \mathrm{{d}}e = T\mathrm{{d}}S + \frac{P}{{{\rho ^2}}}\mathrm{{d}}\rho , \end{aligned}$$

(A.2)

and T is the temperature. Regarding P as a function of $\rho$ and S, $P=P(\rho ,S)$, then the local sound speed c is defined as

$$\begin{aligned} {c^2} = \frac{{\partial P(\rho ,S)}}{{\partial \rho }}. \end{aligned}$$

(A.3)

Thus, the third equation of (A.1) can be replaced equivalently by

$$\begin{aligned} \frac{{\mathrm{{D}}P}}{{\mathrm{{D}}t}} = - \rho {c^2}\frac{{\partial u}}{{\partial x}}. \end{aligned}$$

(A.4)

We observe that the entropy S is constant along a streamline. By the hyperbolic quasilinear analysis for equations (A.1), three eigenvalues of the system are obtained.

$$\begin{aligned} \begin{array}{*{20}{l}} {{\lambda _ - } = u - c},&\quad {{\lambda _0} = u},&\quad {{\lambda _ + } = u + c}. \end{array} \end{aligned}$$

(A.5)

We recall [35], and here introduce the important Riemann invariants $\phi$ and $\psi$ for $\lambda _{-},\lambda _{+}$

$$\begin{aligned} \begin{array}{*{20}{l}} {\phi = u - \mathop \int \limits ^\rho \frac{{c(\omega ,S)}}{\omega }d\omega },\\ {\psi = u + \mathop \int \limits ^\rho \frac{{c(\omega ,S)}}{\omega }d\omega }. \end{array} \end{aligned}$$

(A.6)

We carry out the total differential for (A.6)

$$\begin{aligned} d\psi= & {} du + \frac{c}{\rho }d\rho + \frac{{\partial \psi }}{{\partial S}}dS = du + \frac{1}{{\rho c}}dp + K(\rho ,S)dS,\end{aligned}$$

(A.7)

$$\begin{aligned} d\phi= & {} du - \frac{c}{\rho }d\rho - \frac{{\partial \phi }}{{\partial S}}dS = du - \frac{1}{{\rho c}}dp - K(\rho ,S)dS, \end{aligned}$$

(A.8)

where

$$\begin{aligned} K(\rho ,S) = - \frac{1}{{\rho c}}\frac{{\partial p}}{{\partial S}} + \mathop \int \limits ^\rho \frac{1}{\omega }\frac{{\partial c(\omega ,S)}}{{\partial S}}d\omega , \end{aligned}$$

(A.9)

Along the characteristic ${{C}_{+}}{:}\,dx/dt=u+c$, we have

$$\begin{aligned} \begin{array}{*{20}{l}} {d\psi = K(\rho ,S)dS,}&{dS = c\frac{{\partial S}}{{\partial x}}dt}, \end{array} \end{aligned}$$

(A.10)

and along the characteristic ${{C}_{-}}:dx/dt=u-c$, we have

$$\begin{aligned} \begin{array}{*{20}{l}} {d\phi = - K(\rho ,S)dS,}&\quad {dS = - c\frac{{\partial S}}{{\partial x}}dt}. \end{array} \end{aligned}$$

(A.11)

In particular, when the fluid is polytropic gases, we have

$$\begin{aligned} p = (\gamma - 1)\rho e, \end{aligned}$$

(A.12)

and the sound speed c,

$$\begin{aligned} {c^2} = \frac{{\gamma P}}{\rho }. \end{aligned}$$

(A.13)

The isentropic relation can be written as

$$\begin{aligned} P = {P_0}{\left( {\frac{\rho }{{{\rho _0}}}} \right) ^\gamma }. \end{aligned}$$

(A.14)

In this case, we can get the Riemann invariants as

$$\begin{aligned} \begin{array}{*{20}{l}} {\phi = u - \frac{{2c}}{{\gamma - 1}},}&\quad {\psi = u + \frac{{2c}}{{\gamma - 1}}}. \end{array} \end{aligned}$$

(A.15)

Taking the partial derivative of S for (A.13),

$$\begin{aligned} 2c\frac{{\partial c}}{{\partial S}}= & {} \frac{\gamma }{\rho }\frac{{\partial p}}{{\partial S}},\end{aligned}$$

(A.16)

$$\begin{aligned} \frac{{\partial \psi }}{{\partial S}}= & {} \frac{2}{{(\gamma - 1)}}\frac{{\partial c}}{{\partial S}} = \frac{\gamma }{{(\gamma - 1)\rho c}}\frac{{\partial p}}{{\partial S}}. \end{aligned}$$

(A.17)

Then (A.9) turns to

$$\begin{aligned} K(\rho ,S) = \frac{1}{{(\gamma - 1)\rho c}}\frac{{\partial p}}{{\partial S}} = \frac{T}{c}. \end{aligned}$$

(A.18)

Bringing (A.18) into (A.7) and (A.8)

$$\begin{aligned} d\psi= & {} du + \frac{1}{{\rho c}}dp + \frac{1}{c}TdS,\end{aligned}$$

(A.19)

$$\begin{aligned} d\phi= & {} du - \frac{1}{{\rho c}}dp - \frac{1}{c}TdS, \end{aligned}$$

(A.20)

and the second thermodynamics law can be written as

$$\begin{aligned} TdS = \frac{1}{{(\gamma - 1)\rho }}dp - \frac{{{c^2}}}{{(\gamma - 1)\rho }}d\rho . \end{aligned}$$

(A.21)

Then we have

$$\begin{aligned} TS'= & {} \frac{1}{{(\gamma - 1)\rho }}p' - \frac{{{c^2}}}{{(\gamma - 1)\rho }}\rho ',\end{aligned}$$

(A.22)

$$\begin{aligned} \psi '= & {} u' + \frac{1}{{\rho c}}p' + \frac{1}{c}TS',\end{aligned}$$

(A.23)

$$\begin{aligned} \phi '= & {} u' - \frac{1}{{\rho c}}p' - \frac{1}{c}TS'. \end{aligned}$$

(A.24)

Taking (A.1) to the Riemann invariants and entropy form

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {\frac{{\partial \phi }}{{\partial t}} + (u - c)\frac{{\partial \phi }}{{\partial x}} = T\frac{{\partial S}}{{\partial x}}}\\ {\frac{{\partial \psi }}{{\partial t}} + (u + c)\frac{{\partial \psi }}{{\partial x}} = T\frac{{\partial S}}{{\partial x}}}\\ {\frac{{\partial S}}{{\partial t}} + u\frac{{\partial S}}{{\partial x}} = 0}. \end{array}} \right. \end{aligned}$$

(A.25)

(A.25) still works across the rarefaction wave which is a smooth wave.

If there is a shock (or contact discontinuity) trajectory $r=r(t)$ with speed $\sigma =dr/dt$, assuming ${{U}_{a}},{{U}_{b}}$ are the states on the wave ahead and behind the wave back. Then across the shock, ${{U}_{a}},{{U}_{b}}$ satisfy the Rankine–Hugoniot relations,

$$\begin{aligned} \sigma ({U_a} - {U_b}) = F({U_a}) - F({U_b}). \end{aligned}$$

(A.26)

• (a) When ${{u}_{a}}={{u}_{b}}$ and ${{\rho }_{a}}\ne {{\rho }_{b}}$, the simple wave is a contact discontinuity, ${{U}_{a}},{{U}_{b}}$ satisfy ${{P}_{a}}={{P}_{b}}$, and $\sigma ={{u}_{a}}\pm {{c}_{a}}={{u}_{b}}\pm {{c}_{a}}$

• (b) When ${{u}_{a}}\ne {{u}_{b}}$, the simple wave is a shock wave, from (A.26), we can get

  $$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {\sigma = \frac{{{u_a}{\rho _a} - {u_b}{\rho _b}}}{{{\rho _a} - {\rho _b}}}}\\ {{{({u_a} - {u_b})}^2} = ({P_a} - {P_b})\left( \frac{1}{{{\rho _b}}} - \frac{1}{{{\rho _a}}}\right) }\\ {{e_a} - {e_b} = \frac{{{\rho _a} - {\rho _b}}}{{2{\rho _a}{\rho _b}}}({P_a} + {P_b})}. \end{array}} \right. \end{aligned}$$

  (A.27)

By the combination of EOS function $e=e(\rho ,P)$ and (A.27), we can get the following relation form

$$\begin{aligned}&{u_b} = {u_a} \pm H({p_b};{p_a},{\rho _a}),\end{aligned}$$

(A.28)

$$\begin{aligned}&{\rho _b} = G({P_b};{P_a},{\rho _a}), \end{aligned}$$

(A.29)

here, when the shock is related to ${{\lambda }_{+}}=u+c$, the sign is (+), else should be (−) which is related to ${{\lambda }_{-}}=u-c$.

The partial derivatives are noted as following forms.

$$\begin{aligned}&\begin{array}{*{20}{l}} {{H_1}({p_b};{p_a},{\rho _a}) = \frac{{\partial H}}{{\partial {p_b}}},}&\quad {{H_2}({p_b};{p_a},{\rho _a}) = \frac{{\partial H}}{{\partial {p_a}}}}&\quad {{H_3}({p_b};{p_a},{\rho _a}) = \frac{{\partial H}}{{\partial {\rho _a}}}}, \end{array}\end{aligned}$$

(A.30)

$$\begin{aligned}&\begin{array}{*{20}{l}} {{G_1}({p_b};{p_a},{\rho _a}) = \frac{{\partial G}}{{\partial {p_b}}},}&\quad {{G_2}({p_b};{p_a},{\rho _a}) = \frac{{\partial G}}{{\partial {p_a}}}}&\quad {{G_3}({p_b};{p_a},{\rho _a}) = \frac{{\partial G}}{{\partial {\rho _a}}}}. \end{array} \end{aligned}$$

(A.31)

In particular, when the EOS function is (A.12), taking ${{\mu }^{2}}=(\gamma -1)/(\gamma +1)$, we have

$$\begin{aligned}&H({P_b};{P_a},{\rho _a}) = ({P_b} - {P_a})\sqrt{\frac{{1 - {\mu ^2}}}{{{\rho _a}({P_b} + {\mu ^2}{P_a})}}},\end{aligned}$$

(A.32)

$$\begin{aligned}&\begin{array}{*{20}{l}} {{H_1} = \frac{1}{2}\sqrt{\frac{{1 - {\mu ^2}}}{{{\rho _a}({P_b} + {\mu ^2}{P_a})}}} \frac{{{P_b} + (1 + 2{\mu ^2}){P_a}}}{{{P_b} + {\mu ^2}{P_a}}}}\\ {{H_2} = - \frac{1}{2}\sqrt{\frac{{1 - {\mu ^2}}}{{{\rho _a}({P_b} + {\mu ^2}{P_a})}}} \frac{{(2 + {\mu ^2}){P_b} + {\mu ^2}{P_a}}}{{{P_b} + {\mu ^2}{P_a}}}}\\ {{H_3} = - \frac{{{P_b} - {P_a}}}{{2{\rho _a}}}\sqrt{\frac{{1 - {\mu ^2}}}{{{\rho _a}({P_b} + {\mu ^2}{P_a})}}} }, \end{array}\end{aligned}$$

(A.33)

$$\begin{aligned}&\begin{array}{*{20}{l}} {}&{G({P_b};{P_a},{\rho _a}) = {\rho _a}\frac{{{P_b} + {\mu ^2}{P_a}}}{{{P_a} + {\mu ^2}{P_b}}}}, \end{array}\end{aligned}$$

(A.34)

$$\begin{aligned}&\begin{array}{*{20}{l}} {{G_1} = \frac{{{\rho _a}(1 - {\mu ^4}){P_a}}}{{{{({P_a} + {\mu ^2}{P_b})}^2}}},}&{{G_2} = \frac{{{\rho _a}({\mu ^4} - 1){P_b}}}{{{{({P_a} + {\mu ^2}{P_b})}^2}}},}&{{G_3} = \frac{{{P_b} + {\mu ^2}{P_a}}}{{{P_a} + {\mu ^2}{P_b}}}}. \end{array} \end{aligned}$$

(A.35)

Appendix B: Wave Pattern and Interface Solution of Riemann Problem

Considering the following Riemann problem with the EOS of gas (A.12),

$$\begin{aligned} \begin{array}{*{20}{l}} {\frac{{\partial U}}{{\partial t}} + \frac{{\partial F(U)}}{{\partial x}} = 0}\\ {\begin{array}{*{20}{l}} {U(x,0) = \left\{ {\begin{array}{*{20}{l}} {{U_L}}&{}\quad {x < x_I^0}\\ {{U_R}}&{}\quad {x > x_I^0}. \end{array}} \right. }&{} \end{array}} \end{array} \end{aligned}$$

(B.1)

As shown in Fig. 15, it is known that the wave pattern is three-waves structure [39], which are a left simple wave $W_L$ related to ${{\lambda }_{-}}=u-c$, a contact discontinuity related to ${{\lambda }_{0}}=u$, and a right simple wave $W_R$ related to ${{\lambda }_{+}}=u+c$. The regions between these waves are two constant fluid states $U_{1*}$, $U_{2*}$.

Fig. 15

Wave pattern of Riemann problem

Across the three waves, as shown in Table 1, these states satisfy conditions for Riemann invariants to be equal or satisfy the Rankine–Hugoniot relations.

Table 1 Relations across the waves of Riemann problem

Wave Pattern and Interface Solution of Riemann Problem Obtained by RBC

From the right side state definition (2.6) by RBC, the initial state distributions of Riemann problem (B.1) satisfy

$$\begin{aligned} \begin{array}{*{20}{l}} {{U_L}{:}\,\left\{ {\begin{array}{*{20}{l}} {{u_L}}\\ {{P_L}}\\ {{\rho _L}} \end{array}} \right. },&{}{{U_R}{:}\,\left\{ {\begin{array}{*{20}{l}} {{u_R} = 2u_I^0 - {u_L}}\\ {{P_R} = {P_L}}\\ {{\rho _R} = {\rho _L}} \end{array}} \right. }. \end{array} \end{aligned}$$

(B.2)

Then, we can get the solution of this problem as following form.

Fig. 16

Wave pattern of Riemann problem obtained by RBC when $u_{I}^{0}<{{u}_{L}}$

• a. When $u_{I}^{0}<{{u}_{L}}$, as shown in Fig. 16, the three-waves structure is composed of two shock wave $W_L, W_R$ and one contact discontinuity, and the states $U_{1*},U_{2*}$ satisfy

  $$\begin{aligned}&\begin{array}{*{20}{l}} {{u_{1*}} = {u_L} - H({P_{1*}};{P_L},{\rho _L})}\\ {{\rho _{1*}} = G({P_{1*}};{P_L},{\rho _L})}, \end{array}\end{aligned}$$

  (B.3)
  $$\begin{aligned}&\begin{array}{*{20}{l}} {{u_{2*}} = {u_R} + H({P_{2*}};{P_R},{\rho _R})}\\ {{\rho _{2*}} = G({P_{2*}};{P_R},{\rho _R})}, \end{array}\end{aligned}$$

  (B.4)
  $$\begin{aligned}&\begin{array}{*{20}{l}} {{u_{1*}} = {u_{2*}}}\\ {{P_{1*}} = {P_{2*}}}. \end{array} \end{aligned}$$

  (B.5)

Bringing (B.2) into (B.4)

$$\begin{aligned} \begin{array}{*{20}{l}} {{u_{2*}} = 2u_I^0 - {u_L} + H({P_{2*}};{P_L},{\rho _L})}\\ {{\rho _{2*}} = G({P_{2*}};{P_L},{\rho _L})}. \end{array} \end{aligned}$$

(B.6)

By the combination of (B.5), (B.3) and (B.6), we can get the following relation form

$$\begin{aligned} \begin{array}{*{20}{l}} {{u_{1*}} = {u_{2*}} = u_I^0}\\ {{\rho _{1*}} = {\rho _{2*}} = G({P_{2*}};{P_L},{\rho _L})}. \end{array} \end{aligned}$$

(B.7)

Here, we note the state on the piston interface as $(u_{ shock }^{*},P_{ shock }^{*},\rho _{ shock }^{*}):=(u_{1*},P_{1*},\rho _{1*})$, then they should satisfy

$$\begin{aligned} \begin{array}{*{20}{l}} {u_{ shock }^* = u_I^0}\\ {H(P_{ shock }^*;{P_L},{\rho _L}) = {u_L} - u_{ shock }^*}\\ {\rho _{ shock }^* = G(P_{ shock }^*;{P_L},{\rho _L})}. \end{array} \end{aligned}$$

(B.8)

Fig. 17

Wave pattern of Riemann problem obtained by RBC when $u_{I}^{0}\ge {{u}_{L}}$

• b. When $u_{I}^{0}\ge {{u}_{L}}$, as shown in Fig. 17, the three-waves structure is composed of two centered rarefaction wave $W_L, W_R$ and one contact discontinuity, and the states $U_{1*},U_{2*}$ satisfy

  $$\begin{aligned}&\begin{array}{*{20}{l}} {\int \limits _{{U_L}}^{{U_{1*}}} {d\psi } = {u_{1*}} - {u_L} + \int \limits _{{P_L}}^{{P_{1*}}} {\frac{1}{{\rho c}}dP} = 0}\\ {S({\rho _{1*}},{P_{1*}}) = S({\rho _L},{P_L})}, \end{array}\end{aligned}$$

  (B.9)
  $$\begin{aligned}&\begin{array}{*{20}{l}} {\int \limits _{{U_R}}^{{U_{2*}}} {d\phi } = {u_{2*}} - {u_R} - \int \limits _{{P_R}}^{{P_{2*}}} {\frac{1}{{\rho c}}dP} = 0}\\ {S({\rho _{2*}},{P_{2*}}) = S({\rho _R},{P_R})}, \end{array}\end{aligned}$$

  (B.10)
  $$\begin{aligned}&\begin{array}{*{20}{l}} {{u_{1*}} = {u_{2*}}}\\ {{P_{1*}} = {P_{2*}}}. \end{array} \end{aligned}$$

  (B.11)

Bringing (B.2) into (B.10)

$$\begin{aligned} \begin{array}{*{20}{l}} {{u_{2*}} = 2u_I^0 - {u_L} + \int \limits _{{P_L}}^{{P_{2*}}} {\frac{1}{{\rho c}}dP} = 0}\\ {S({\rho _{2*}},{P_{2*}}) = S({\rho _L},{P_L})}. \end{array} \end{aligned}$$

(B.12)

By the combination of (B.9), (B.11) and (B.12), we can get the following relation form

$$\begin{aligned} \begin{array}{*{20}{l}} {{u_{1*}} = {u_{2*}} = u_I^0}\\ {{\rho _{1*}} = {\rho _{2*}}}. \end{array} \end{aligned}$$

(B.13)

Here, we note the state on the piston interface as $(u_{ rare }^{*},P_{ rare }^{*},\rho _{ rare }^{*}):=({{u}_{1*}},{{P}_{1*}},{{\rho }_{1*}})$, then they should satisfy

$$\begin{aligned} \begin{array}{*{20}{l}} {u_{ rare }^* = u_I^0},\\ {\int \limits _{{P_L}}^{P_{ rare }^*} {\frac{1}{{\rho c}}dP} = {u_L} - u_{ rare }^*},\\ {S(\rho _{ rare }^*,P_{ rare }^*) = S({\rho _L},{P_L})}. \end{array} \end{aligned}$$

(B.14)

Appendix C: Wave Pattern and Interface Solution of Generalized Riemann Problem

Considering the following generalized Riemann problem,

$$\begin{aligned} \begin{array}{*{20}{l}} {\frac{{\partial U}}{{\partial t}} + \frac{{\partial F(U)}}{{\partial x}} = 0}\\ {\begin{array}{*{20}{l}} {U(x,0) = \left\{ {\begin{array}{*{20}{l}} {{U_L} + (x - x_I^0){{U'}_L}}&{}\quad {x < x_I^0}\\ {{U_R} + (x - x_I^0){{U'}_R}}&{}\quad {x > x_I^0}. \end{array}} \right. }&{} \end{array}} \end{array} \end{aligned}$$

(C.1)

As shown in Fig. 18, The solution is piecewise smooth in a small-time range.

Note ${{U}_{1*}}={\mathop {\lim }\nolimits _{t\rightarrow 0+}}\,{{U}_{1}}({{x}_{I}}-,t)$ and ${{U}_{2*}}={\mathop {\lim }\nolimits _{t\rightarrow 0+}}\,{{U}_{2}}({{x}_{I}}+,t)$, referring to [35], it is known that $(U_{1*}^{{}},U_{2*}^{{}})$ are the solutions on both side of the contact discontinuity for Riemann problem (B.1). With time $t\rightarrow 0+$ and $x\rightarrow {{x}_{I}}-$, the limiting material derivatives $(Du/Dt)_{1*}$, ${{(DP/Dt)}_{1*}}$ behind the left wave $W_L$ satisfy

$$\begin{aligned} {A_L}\left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - + {B_L}\left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = {D_L}, \end{aligned}$$

(C.2)

Fig. 18

Wave pattern of generalized Riemann problem

with

$$\begin{aligned}&({A_L},{B_L},{D_L}) = \left\{ {\begin{array}{*{20}{ll}} {(A_L^{ shock },B_L^{ shock },D_L^{ shock })}&{}\quad {if}{{u_L} > {u_{1*}}}\\ {(A_L^{ Rare },B_L^{ Rare },D_L^{ Rare })}&{}\quad {if}{{u_L} \le {u_{1*}}}, \end{array}} \right. \end{aligned}$$

(C.3)

$$\begin{aligned}&\begin{array}{*{20}{l}} {A_L^{ shock } = 1 - {\rho _{1*}}({\sigma _L} - {u_{1*}})H_1^L},&{}{B_L^{ shock } = - \left[ {\frac{1}{{{\rho _{1*}}c_{1*}^2}}({\sigma _L} - {u_{1*}}) - H_1^L} \right] },\\ {D_L^{ shock } = L_P^L{{P'}_L} + L_u^L{{u'}_L} + L_\rho ^L{{\rho '}_L}}, &{} {{\sigma _L} = \frac{{{\rho _L}{u_L} - {\rho _{1*}}{u_{1*}}}}{{{\rho _L} - {\rho _{1*}}}}}, \end{array}\end{aligned}$$

(C.4)

$$\begin{aligned}&\begin{array}{*{20}{l}} {L_P^L = - \frac{1}{{{\rho _L}}} - ({\sigma _L} - {u_L})H_2^L,}&{}{}&{}{L_u^L = {\sigma _L} - {u_L} + ({\rho _L}c_L^2H_2^L + {\rho _L}H_3^L),}\\ {L_\rho ^L = - ({\sigma _L} - {u_L})H_3^L.}&{}{}&{}{} \end{array} \end{aligned}$$

(C.5)

Here, $H_{i}^{L}={{H}_{i}}({{P}_{*}};{{P}_{L}},{{\rho }_{L}}),i=1,2,3$ are given in (A.30) and (A.33).

$$\begin{aligned}&\begin{array}{*{20}{l}} {A_L^{ rare } = 1,}&{B_L^{ rare } = \frac{1}{{{\rho _{1*}}{c_{1*}}}}}, \end{array}\end{aligned}$$

(C.6)

$$\begin{aligned}&D_L^{ rare } \!= \! \left[ {\frac{{1 \!+\! {\mu ^2}}}{{1 \!+\! 2{\mu ^2}}}{{\left( {\frac{{{c_{1*}}}}{{{c_L}}}} \right) }^{1/(2{\mu ^2})}} \!+\! \frac{{{\mu ^2}}}{{1 \!+\! 2{\mu ^2}}}{{\left( {\frac{{{c_{1*}}}}{{{c_L}}}} \right) }^{(1 + {\mu ^2})/{\mu ^2}}}} \right] {T_L}{S'_L} - {c_L}{\left( {\frac{{{c_{1*}}}}{{{c_L}}}} \right) ^{1/(2{\mu ^2})}}{\psi '_L}.\nonumber \\ \end{aligned}$$

(C.7)

Here, ${{T}_{L}}{{{S}'}_{L}}$, ${{{\psi }'}_{L}}$ are given in (A.22) and (A.23).

Particularly, when ${{u}_{*}}={{u}_{L}}$, (C.7) becomes

$$\begin{aligned} D_L^{ rare } = - \frac{1}{{{\rho _L}}}{P'_L} - {c_L}{u'_L}. \end{aligned}$$

(C.8)

The limiting material derivatives ${{(Du/Dt)}_{2*}}$, ${{(DP/Dt)}_{2*}}$ behind the right wave ${{W}_{R}}$ satisfy

$$\begin{aligned} {A_R}\left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + + {B_R}\left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + = {D_R}, \end{aligned}$$

(C.9)

with

$$\begin{aligned}&({A_R},{B_R},{D_R}) = \left\{ {\begin{array}{*{20}{ll}} {(A_R^{ shock },B_R^{ shock },D_R^{ shock })}&{}\quad {if}{{u_{2*}} > {u_R}}\\ {(A_R^{ Rare },B_R^{ Rare },D_R^{ Rare })}&{}\quad {if}{{u_{2*}} \le {u_R}}, \end{array}} \right. \end{aligned}$$

(C.10)

$$\begin{aligned}&\begin{array}{*{20}{l}} {A_R^{ shock } = 1 + {\rho _{2*}}({\sigma _R} - {u_*})H_1^R},&{}{B_R^{ shock } = - \left[ {\frac{1}{{{\rho _{2*}}c_{2*}^2}}({\sigma _R} - {u_*}) + H_1^R} \right] },\\ {D_R^{ shock } = L_P^R{{P'}_R} + L_u^R{{u'}_R} + L_\rho ^R{{\rho '}_R}},&{}{{\sigma _R} = \frac{{{\rho _R}{u_R} - {\rho _{2*}}{u_{2*}}}}{{{\rho _R} - {\rho _{2*}}}}}, \end{array}\nonumber \\ \end{aligned}$$

(C.11)

$$\begin{aligned}&\begin{array}{*{20}{l}} {L_P^R = - \frac{1}{{{\rho _R}}} + ({\sigma _R} - {u_R})H_2^R,}&{}{}&{}{L_u^R = {\sigma _R} - {u_R} - ({\rho _R}c_R^2H_2^R + {\rho _R}H_3^R),}\\ {L_\rho ^R = ({\sigma _R} - {u_R})H_3^R.}&{}{}&{}{} \end{array} \end{aligned}$$

(C.12)

Here, $H_{i}^{R}={{H}_{i}}({{P}_{*}};{{P}_{R}},{{\rho }_{R}}),i=1,2,3$, are given in (A.30) and (A.33).

$$\begin{aligned}&\left( {A_R^{ rare },B_R^{ rare }} \right) = \left( {1, - \frac{1}{{{\rho _{2*}}{c_{2*}}}}} \right) ,\end{aligned}$$

(C.13)

$$\begin{aligned}&{D_{ rare }} \!= \!\left[ {\frac{{1 \!+\! {\mu ^2}}}{{1 \!+\! 2{\mu ^2}}}{{\left( {\frac{{{c_{2*}}}}{{{c_R}}}} \right) }^{1/(2{\mu ^2})}} \!+\! \frac{{{\mu ^2}}}{{1 + 2{\mu ^2}}}{{\left( {\frac{{{c_{2*}}}}{{{c_R}}}} \right) }^{(1 \!+\! {\mu ^2})/{\mu ^2}}}} \right] {T_R}{S'_R} + {c_R}{\left( {\frac{{{c_{2*}}}}{{{c_R}}}} \right) ^{1/(2{\mu ^2})}}{\phi '_R}.\nonumber \\ \end{aligned}$$

(C.14)

Here, ${{T}_{R}}{{{S}'}_{R}}$, ${{{\phi }'}_{R}}$ are given in (A.22) and (A.24).

Particularly, when ${{u}_{*}}={{u}_{R}}$, (C.14) turns to

$$\begin{aligned} D_R^{ rare } = - \frac{1}{{{\rho _R}}}{P'_R} + {c_R}{u'_R}. \end{aligned}$$

(C.15)

Thus, turning the material derivatives to spatial derivatives, we have the limiting spatial derivatives ${{(\partial u/\partial x)}_{1*}}$, ${{(\partial \rho /\partial x)}_{1*}}$, ${{(\partial P/\partial x)}_{1*}}$ behind the left wave $W_L$ satisfy

$$\begin{aligned} \begin{array}{*{20}{l}} {\frac{{\partial {P_{1*}}}}{{\partial x}} = - \rho _{1*}^{}\frac{{D{u_{1*}}}}{{Dt}}},\\ {\frac{{\partial {u_{1*}}}}{{\partial x}} = - \frac{1}{{\rho _{1*}^{}c_{1*}^2}}\frac{{D{P_{1*}}}}{{Dt}}},\\ {\frac{{\partial {\rho _{1*}}}}{{\partial x}} = \left\{ {\begin{array}{*{20}{l}} {\frac{{\partial \rho _{ shock }^{1*}}}{{\partial x}}}&{}\quad {u_*^{} < {u_L}}\\ {\frac{{\partial \rho _{ rare }^{1*}}}{{\partial x}}}&{}\quad {u_*^{} \ge {u_L}} \end{array}} \right. }. \end{array} \end{aligned}$$

(C.16)

Here, $\frac{\partial \rho _{ shock }^{1*}}{\partial x}$ and $\frac{\partial \rho _{ rare }^{1*}}{\partial x}$ satisfy

$$\begin{aligned}&\frac{{\partial {P_{1*}}}}{{\partial x}} - c_{1*}^2\frac{{\partial \rho _{ rare }^{1*}}}{{\partial x}} = (\gamma - 1){\rho _{1*}}{\left( {\frac{{{c_{1*}}}}{{{c_L}}}} \right) ^{\frac{{1 + {\mu ^2}}}{{{\mu ^2}}}}} \cdot {T_L}S{'_L},\end{aligned}$$

(C.17)

$$\begin{aligned}&g_{\rho ,L}^{ shock }\frac{{\partial {\rho _{1*}}}}{{\partial x}} + g_{P,L}^{ shock }\frac{{\partial {P_{1*}}}}{{\partial x}} + g_{u,L}^{ shock }\frac{{\partial {u_{1*}}}}{{\partial x}} = f_L^{ shock }, \end{aligned}$$

(C.18)

with

$$\begin{aligned}&{g_{\rho ,L}^{ shock } = {\sigma _L} - {u_{1*}},}\quad {g_{P,L}^{ shock } = - ({\sigma _L} - {u_{1*}}){G_1},}\quad {g_{u,L}^{ shock } = - {\rho _{1*}} + {\rho _{1*}}c_{1*}^2{G_1}},\nonumber \\&f_L^{ shock } = ({\sigma _L} - {u_L}){G_2}{P'_L} + ({\sigma _L} - {u_L}){G_3}{\rho '_L} - {\rho _L}({G_2}c_L^2 + {G_3}){u'_L}. \end{aligned}$$

(C.19)

where ${{G}_{i}}$, $i = 1,2,3$ are given in (A.35);

The limiting spatial derivatives ${{(\partial u/\partial x)}_{2*}}$, ${{(\partial \rho /\partial x)}_{2*}}$, ${{(\partial P/\partial x)}_{2*}}$ behind the right wave ${{W}_{R}}$ satisfy

$$\begin{aligned} \begin{array}{*{20}{l}} {\frac{{\partial {P_{2*}}}}{{\partial x}} = - \rho _{2*}^{}\frac{{D{u_{2*}}}}{{Dt}}},\\ {\frac{{\partial {u_{2*}}}}{{\partial x}} = - \frac{1}{{\rho _{2*}^{}c_{2*}^2}}\frac{{D{P_{2*}}}}{{Dt}}},\\ {\frac{{\partial {\rho _{2*}}}}{{\partial x}} = \left\{ {\begin{array}{*{20}{l}} {\frac{{\partial \rho _{ shock }^{2*}}}{{\partial x}}}&{}\quad {u_*^{} > {u_R}}\\ {\frac{{\partial \rho _{ rare }^{2*}}}{{\partial x}}}&{}\quad {u_*^{} \le {u_R}} \end{array}} \right. }. \end{array} \end{aligned}$$

(C.20)

Here, $\frac{\partial \rho _{ shock }^{2*}}{\partial x}$ and $\frac{\partial \rho _{ rare }^{2*}}{\partial x}$ satisfy

$$\begin{aligned}&\frac{{\partial {P_{2*}}}}{{\partial x}} - c_{2*}^2\frac{{\partial \rho _{ rare }^{2*}}}{{\partial x}} = (\gamma - 1){\rho _{2*}}{\left( {\frac{{{c_{2*}}}}{{{c_R}}}} \right) ^{\frac{{1 + {\mu ^2}}}{{{\mu ^2}}}}} \cdot {T_R}S{'_R},\end{aligned}$$

(C.21)

$$\begin{aligned}&g_{\rho ,R}^{ shock }\frac{{\partial {\rho _{2*}}}}{{\partial x}} + g_{P,R}^{ shock }\frac{{\partial {P_{2*}}}}{{\partial x}} + g_{u,R}^{ shock }\frac{{\partial {u_{2*}}}}{{\partial x}} = f_R^{ shock }, \end{aligned}$$

(C.22)

with

$$\begin{aligned}&{g_{\rho ,R}^{ shock } = {\sigma _R} - {u_{2*}},}\quad {g_{P,R}^{ shock } = - ({\sigma _R} - {u_{2*}}){G_1},}\quad {g_{u,R}^{ shock } = - {\rho _{2*}} + {\rho _{2*}}c_{2*}^2{G_1},}\nonumber \\&f_R^{ shock } = ({\sigma _R} - {u_R}){G_2}{P'_R} + ({\sigma _R} - {u_R}){G_3}{\rho '_R} - {\rho _R}({G_2}c_R^2 + {G_3}){u'_R}. \end{aligned}$$

(C.23)

Here ${{G}_{i}}$, $i = 1,2,3$ are given in (A.35).

Wave Pattern and Interface Solution of Generalized Riemann Problem Obtained by RBC

From the right side state definition (2.6) by RBC, the initial state distributions of generalized Riemann problem (C.1) satisfy

$$\begin{aligned} \begin{array}{*{20}{l}} {{U_L}{:}\,\left\{ {\begin{array}{*{20}{l}} {{u_L}}\\ {{P_L}}\\ {{\rho _L}} \end{array}} \right. },&{}\quad {{U_R}{:}\,\left\{ {\begin{array}{*{20}{l}} {{u_R} = 2u_I^0 - {u_L}}\\ {{P_R} = {P_L}}\\ {{\rho _R} = {\rho _L}} \end{array}} \right. },\\ {{{U'}_L}{:}\,\left\{ {\begin{array}{*{20}{l}} {{{u'}_L}}\\ {{{P'}_L}}\\ {{{\rho '}_L}} \end{array}} \right. },&{}\quad {{{U'}_R}{:}\,\left\{ {\begin{array}{*{20}{l}} {{{u'}_R} = {{u'}_L}}\\ {{{P'}_R} = - {{P'}_L}}\\ {{{\rho '}_R} = - {{\rho '}_L}} \end{array}} \right. }. \end{array} \end{aligned}$$

(C.24)

Then, as shown in Fig. 19, we can get the solution of this problem as following form.

Fig. 19

Wave pattern of generalized Riemann problem obtained by RBC

• a. When $u_{I}^{0}<{{u}_{L}}$, the three-waves structure is composed of two shock wave $W_L, W_R$ and one contact discontinuity, the limiting states ${{U}_{1*}}$, ${{U}_{2*}}$ and their material derivatives ${{(Du/Dt)}_{1*}}$, ${{(DP/Dt)}_{1*}}$, ${{(Du/Dt)}_{2*}}$, ${{(DP/Dt)}_{2*}}$ behind the wave $W_L, W_R$ satisfy

  $$\begin{aligned}&{U_{1*}} = {U_{2*}},\end{aligned}$$

  (C.25)
  $$\begin{aligned}&A_L^{ shock }\left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - + B_L^{ shock }\left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = D_L^{ shock },\end{aligned}$$

  (C.26)
  $$\begin{aligned}&A_R^{ shock }\left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + + B_R^{ shock }\left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + = D_R^{ shock },\end{aligned}$$

  (C.27)
  $$\begin{aligned}&\begin{array}{*{20}{l}} {\left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - = \left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + },\\ {\left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = \left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + }. \end{array} \end{aligned}$$

  (C.28)

Here, ${{U}_{1*}}$, ${{U}_{2*}}$ are given in (B.8).

Bringing (C.24), (C.25) into (C.4) and (C.11), we can get

$$\begin{aligned} \begin{array}{*{20}{l}} {{\sigma _R} = 2u_I^0 - {\sigma _L}},\\ {A_R^{ shock } = A_L^{ shock }},\\ {B_R^{ shock } = - B_L^{ shock }},\\ {D_R^{ shock } = - D_L^{ shock }}. \end{array} \end{aligned}$$

(C.29)

Then, (C.27) becomes

$$\begin{aligned} A_L^{ shock }\left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + - B_L^{ shock }\left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + = - D_L^{ shock }. \end{aligned}$$

(C.30)

From (C.28), (C.26) and (C.30), we have

$$\begin{aligned} \begin{array}{*{20}{l}} {\left( {\frac{{Du}}{{Dt}}} \right) _*^{}: = \left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - = 0},\\ {\left( {\frac{{DP}}{{Dt}}} \right) _*^{}: = \left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = \frac{{D_L^{ shock }}}{{B_L^{ shock }}}}. \end{array} \end{aligned}$$

(C.31)

• b. When $u_{I}^{0}\ge {{u}_{L}}$, the three-waves structure is composed of two rarefaction wave $W_L, W_R$ and one contact discontinuity, the limiting states ${{U}_{1*}}$, ${{U}_{2*}}$ and their material derivatives ${{(Du/Dt)}_{1*}}$, ${{(DP/Dt)}_{1*}}$, ${{(Du/Dt)}_{2*}}$, ${{(DP/Dt)}_{2*}}$ behind the wave $W_L, W_R$ satisfy

  $$\begin{aligned}&{U_{1*}} = {U_{2*}},\end{aligned}$$

  (C.32)
  $$\begin{aligned}&A_L^{ rare }\left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - + B_L^{ rare }\left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = D_L^{ rare },\end{aligned}$$

  (C.33)
  $$\begin{aligned}&A_R^{ rare }\left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + + B_R^{ rare }\left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + = D_R^{ rare },\end{aligned}$$

  (C.34)
  $$\begin{aligned}&\begin{array}{*{20}{l}} {\left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - = \left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + },\\ {\left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = \left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + }. \end{array} \end{aligned}$$

  (C.35)

Here, ${{U}_{1*}}$, ${{U}_{2*}}$ are given in (B.14).

Bringing (C.24), (C.32) into (C.6) and (C.13), we can get

$$\begin{aligned} \begin{array}{*{20}{l}} {A_R^{ rare } = A_L^{ rare }},\\ {B_R^{ rare } = - B_L^{ rare }},\\ {D_R^{ rare } = - D_L^{ rare }}. \end{array} \end{aligned}$$

(C.36)

Then, (C.34) becomes

$$\begin{aligned} A_L^{ rare }\left( {\frac{{Du}}{{Dt}}} \right) _{2*}^ + - B_L^{ rare }\left( {\frac{{DP}}{{Dt}}} \right) _{2*}^ + = - D_L^{ rare }. \end{aligned}$$

(C.37)

From (C.33), (C.37) and (C.35), we have

$$\begin{aligned}&\left( {\frac{{Du}}{{Dt}}} \right) _*^{}: = \left( {\frac{{Du}}{{Dt}}} \right) _{1*}^ - = 0,\end{aligned}$$

(C.38)

$$\begin{aligned}&\left( {\frac{{DP}}{{Dt}}} \right) _*^{}: = \left( {\frac{{DP}}{{Dt}}} \right) _{1*}^ - = \frac{{D_L^{ rare }}}{{B_L^{ rare }}}. \end{aligned}$$

(C.39)

Appendix D: Codes for PRP-Solver and GPRP-Solver