A Quasi-Conservative Discontinuous Galerkin Method for Solving Five Equation Model of Compressible Two-Medium Flows

Abstract

In this work, we develop a quasi-conservative discontinuous Galerkin method for the simulation of compressible gas-gas and gas-water two-medium flows by solving the five-equation transport model. This spatial discretization is a direct extension of the quasi-conservative finite volume discretization to the discontinuous Galerkin framework, thus, preserves uniform velocity and pressure fields at an isolated material interface. Furthermore, for discontinuities with a large pressure ratio, low density, and a dramatic change of material property where nonphysical values may occur, a strategy for imposing the bound-preserving limiting for volume fraction and a positivity-preserving limiting for density of each fluid and internal energy is developed and analyzed based on the quasi-conservative DG($p_1$) discretization. Typical test cases for both one- and two-dimensional problems are provided to demonstrate the performance of the proposed method.

Appendix A

In this appendix, we briefly describe the derivation of the quasi-conservative DG spatial discretization for the five-equation transport model. In order to derive the spatial discretization, we multiply the governing equations Eq. (1) with a basis function $\phi _k$ on a given cell $I_j$, perform an integration by parts, and we have

$$\begin{aligned} \begin{aligned}&\int _{I_j} \frac{\partial (z_1\rho _1)_h}{\partial t} \phi _k \mathrm {d}x + \widehat{(z_1\rho _1)_hu_h} \phi _k\Big |_{j+\frac{1}{2}} - \widehat{(z_1\rho _1)_hu_h} \phi _k\Big |_{j-\frac{1}{2}} - \int _{I_j} (z_1\rho _1)_h u_h\frac{\partial \phi _k}{\partial x}\mathrm {d}x = 0,\\&\int _{I_j} \frac{\partial (z_2\rho _2)_h}{\partial t} \phi _k \mathrm {d}x + \widehat{(z_2\rho _2)_hu_h} \phi _k\Big |_{j+\frac{1}{2}} - \widehat{(z_2\rho _2)_hu_h} \phi _k\Big |_{j-\frac{1}{2}} - \int _{I_j} (z_2\rho _2)_h u_h\frac{\partial \phi _k}{\partial x}\mathrm {d}x = 0,\\&\int _{I_j} \frac{\partial \rho u_h}{\partial t} \phi _k \mathrm {d}x + \widehat{(\rho _h u_h^2 + p_h)} \phi _k\Big |_{j+\frac{1}{2}} - \widehat{(\rho _h u_h^2 + p_h)} \phi _k\Big |_{j-\frac{1}{2}} \\&\quad - \int _{I_j} (\rho _h u_h^2+p_h) \frac{\partial \phi _k}{\partial x}\mathrm {d}x = 0,\\&\int _{I_j} \frac{\partial E_h}{\partial t} \phi _k \mathrm {d}x + \widehat{u_h(E_h+p_h)} \phi _k\Big |_{j+\frac{1}{2}} - \widehat{u_h(E_h+p_h)} \phi _k\Big |_{j-\frac{1}{2}} \\&\quad - \int _{I_j} u_h(E_h+p_h) \frac{\partial \phi _k}{\partial x}\mathrm {d}x = 0,\\&\int _{I_j} \frac{\partial (z_1)_h}{\partial t} \phi _k \mathrm {d}x + \widehat{u_h(z_1)_h} \phi _k\Big |_{j+\frac{1}{2}} - \widehat{u_h(z_1)_h} \phi _k\Big |_{j-\frac{1}{2}} + \int _{I_j} (z_1)_h \frac{\partial u_h\phi _k}{\partial x}\mathrm {d}x = 0,\\ \end{aligned} \end{aligned}$$

(78)

where $(\widehat{(z_1\rho _1)_hu_h},\widehat{(z_2\rho _2)_hu_h},\widehat{\rho _h u_h^2 + p_h},\widehat{u_h(E_h+p_h)},\widehat{u_h(z_1)_h})^\top \Big |_{j\pm \frac{1}{2}}$ are the numerical flux vectors at the cell interface $x=x_{j\pm \frac{1}{2}}$, respectively, and will be specified later.

As we would like to remove the spatial derivatives with respective to the velocity $u_h$ in the last cell integral in Eq. (78), therefore, an extra integration by parts is performed locally in the cell $I_j$, then we obtain

$$\begin{aligned} \begin{aligned}&\int _{I_j} \frac{\partial (z_1\rho _1)_h}{\partial t} \phi _k \mathrm {d}x + \widehat{(z_1\rho _1)_hu_h} \phi _k\Big |_{j+\frac{1}{2}} - \widehat{(z_1\rho _1)_hu_h} \phi _k\Big |_{j-\frac{1}{2}} - \int _{I_j} (z_1\rho _1)_h u_h\frac{\partial \phi _k}{\partial x}\mathrm {d}x = 0,\\&\int _{I_j} \frac{\partial (z_2\rho _2)_h}{\partial t} \phi _k \mathrm {d}x + \widehat{(z_2\rho _2)_hu_h} \phi _k\Big |_{j+\frac{1}{2}} - \widehat{(z_2\rho _2)_hu_h} \phi _k\Big |_{j-\frac{1}{2}} - \int _{I_j} (z_2\rho _2)_h u_h\frac{\partial \phi _k}{\partial x}\mathrm {d}x = 0,\\&\int _{I_j} \frac{\partial \rho u_h}{\partial t} \phi _k \mathrm {d}x + \widehat{(\rho _h u_h^2 + p_h)} \phi _k\Big |_{j+\frac{1}{2}} - \widehat{(\rho _h u_h^2 + p_h)} \phi _k\Big |_{j-\frac{1}{2}} \\&\quad - \int _{I_j} (\rho _h u_h^2+p_h) \frac{\partial \phi _k}{\partial x}\mathrm {d}x = 0,\\&\int _{I_j} \frac{\partial E_h}{\partial t} \phi _k \mathrm {d}x + \widehat{u_h(E_h+p_h)} \phi _k\Big |_{j+\frac{1}{2}} - \widehat{u_h(E_h+p_h)} \phi _k\Big |_{j-\frac{1}{2}} \\&\quad - \int _{I_j} u_h(E_h+p_h) \frac{\partial \phi _k}{\partial x}\mathrm {d}x = 0,\\&\int _{I_j} \frac{\partial (z_1)_h}{\partial t} \phi _k \mathrm {d}x + \widehat{u_h(z_1)_h} \phi _k\Big |_{j+\frac{1}{2}} - \widehat{u_h(z_1)_h} \phi _k\Big |_{j-\frac{1}{2}} + \int _{I_j} u_h \frac{\partial (z_1)_h}{\partial x}\phi _k\mathrm {d}x\\&\quad -[(u_h(z_1)_h \phi _k)^-_{j+\frac{1}{2}} - (u_h(z_1)_h \phi _k)^+_{j-\frac{1}{2}}]= 0,\\ \end{aligned} \end{aligned}$$

(79)

We note that the second integration by parts is different from the first integration by parts, as it is performed on the discretized level within the cell $I_j$, thus, no numerical flux is involved.

Next, we need to further specify the formulations of the numerical flux. For the first four conservative variables, we use the Lax-Friedrichs numerical flux, which is given as

$$\begin{aligned} \widehat{\varvec{F}}(\varvec{u}_{j+\frac{1}{2}}^-,\varvec{u}_{j+\frac{1}{2}}^+) = \frac{1}{2}[\varvec{F}(\varvec{u}_{j+\frac{1}{2}}^+)+\varvec{F}(\varvec{u}_{j+\frac{1}{2}}^-) - S_{j+\frac{1}{2}}(\varvec{u}_{j+\frac{1}{2}}^+ - \varvec{u}_{j+\frac{1}{2}}^-)], \end{aligned}$$

(80)

where $\varvec{u}=(z_1\rho _1, z_2\rho _2, \rho u, E)^\top$ and $\varvec{F}(\varvec{u})=(z_1\rho _1u, z_2\rho _2u, \rho u^2+p, u(E+p))^\top$. To further derive the numerical flux for the equation of volume fraction, we consider an isolated interface problem, which contains two components with different densities and volume fractions moving at a uniform velocity and a uniform pressure. We assume that the isolated interface is located at the cell $I_j$.

After substituting the Lax-Friedrichs numerical flux into the first four discretized equations in Eq. (79), we obtain a discretized equation for the variable $z_1$ as follows

$$\begin{aligned} \begin{aligned}&\int _{I_j} \frac{\partial (z_1)_h}{\partial t} \phi _k \mathrm {d}x +\frac{1}{2}\left[ u(z_1)_{j+\frac{1}{2}}^+ + u(z_1)_{j+\frac{1}{2}}^- - S_{j+\frac{1}{2}}((z_1)_{j+\frac{1}{2}}^+-(z_1)_{j+\frac{1}{2}}^-)\right] \phi _k|^-_{j+\frac{1}{2}}\\&\quad -\frac{1}{2}\left[ u(z_1)_{j-\frac{1}{2}}^+ + u(z_1)_{j-\frac{1}{2}}^- - S_{j-\frac{1}{2}}((z_1)_{j-\frac{1}{2}}^+-(z_1)_{j-\frac{1}{2}}^-)\right] \phi _k|^+_{j-\frac{1}{2}}\\&\quad -\int _{I_j} u(z_1)_h\frac{\partial \phi _k}{\partial x}\mathrm {d}x= 0. \end{aligned} \end{aligned}$$

(81)

An extra integral by parts is performed locally in the cell $I_j$ and since u is uniform at time $t^n$, we obtain

$$\begin{aligned} \begin{aligned}&\int _{I_j} \frac{\partial (z_1)_h}{\partial t} \phi _k \mathrm {d}x +\frac{1}{2}\left[ u(z_1)_{j+\frac{1}{2}}^+ + u(z_1)_{j+\frac{1}{2}}^- - S_{j+\frac{1}{2}}((z_1)_{j+\frac{1}{2}}^+-(z_1)_{j+\frac{1}{2}}^-)\right] \phi _k|^-_{j+\frac{1}{2}}\\&\quad -\frac{1}{2}\left[ u(z_1)_{j-\frac{1}{2}}^+ + u(z_1)_{j-\frac{1}{2}}^- - S_{j-\frac{1}{2}}\left( (z_1)_{j-\frac{1}{2}}^+-(z_1)_{j-\frac{1}{2}}^-\right) \right] \phi _k|^+_{j-\frac{1}{2}}\\&\quad +\int _{I_j} u\frac{\partial (z_1)_h}{\partial x}\phi _k\mathrm {d}x-\left[ (u(z_1)_h\phi _k)^-_{j+\frac{1}{2}} - (u(z_1)_h \phi _k)^+_{j-\frac{1}{2}}\right] = 0. \end{aligned} \end{aligned}$$

(82)

We compare the above spatial discretization for the variable $z_1$ with the discretization given in Eq. (79), which leads to

$$\begin{aligned} \widehat{u(z_1)}_h \Big |_{j\pm \frac{1}{2}} = \frac{1}{2}\left[ u(z_1)_{j\pm \frac{1}{2}}^+ + u(z_1)_{j\pm \frac{1}{2}}^- - S_{j\pm \frac{1}{2}}\left( (z_1)_{j\pm \frac{1}{2}}^+-(z_1)_{j\pm \frac{1}{2}}^-\right) \right] , \end{aligned}$$

(83)

where the subscript and superscript of u in above formula need to be specified further.

Finally, in order to properly define the velocity, we require that in Eq. (82) the residual of the spatial discretization equals to zero for single fluid flows. This consistent requirement leads to the final numerical flux as follows

$$\begin{aligned}&\widehat{u_h(z_1)_h}\Big |_{j+\frac{1}{2}}\nonumber \\&\quad = {\left\{ \begin{array}{ll} \frac{1}{2}\left[ u_{j+\frac{1}{2}}^-(z_1)_{j+\frac{1}{2}}^++u_{j+\frac{1}{2}}^-(z_1)_{j+\frac{1}{2}}^- - S_{j+\frac{1}{2}}\left( (z_1)_{j+\frac{1}{2}}^+-(z_1)_{j+\frac{1}{2}}^-\right) \right] \quad &{}\text {for}\quad I_j,\\ \frac{1}{2}\left[ u_{j+\frac{1}{2}}^+(z_1)_{j+\frac{1}{2}}^++u_{j+\frac{1}{2}}^+(z_1)_{j+\frac{1}{2}}^- - S_{j+\frac{1}{2}}\left( (z_1)_{j+\frac{1}{2}}^+-(z_1)_{j+\frac{1}{2}}^-\right) \right] \quad &{}\text {for}\quad I_{j+1}.\\ \end{array}\right. } \end{aligned}$$

(84)

The formulation of the numerical flux at $x=x_{j-\frac{1}{2}}$ is similar, thus, omitted.