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A Quasi-Conservative Discontinuous Galerkin Method for Multi-component Flows Using the Non-oscillatory Kinetic Flux

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Abstract

In this paper, a high order quasi-conservative discontinuous Galerkin (DG) method using the non-oscillatory kinetic flux is proposed for the 5-equation model of compressible multi-component flows with Mie-Grüneisen equation of state. The method mainly consists of three steps: firstly, the DG method with the non-oscillatory kinetic flux is used to solve the conservative equations of the model; secondly, inspired by Abgrall’s idea, we derive a DG scheme for the volume fraction equation which can avoid the unphysical oscillations near the material interfaces; finally, a multi-resolution weighted essentially non-oscillatory limiter and a maximum-principle-satisfying limiter are employed to ensure oscillation-free near the discontinuities, and preserve the physical bounds for the volume fraction, respectively. Numerical tests show that the method can achieve high order for smooth solutions and keep non-oscillatory at discontinuities. Moreover, the velocity and pressure are oscillation-free at the interface and the volume fraction can stay in the interval [0,1].

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Dongmi Luo & Yibing Chen

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, China

    Jianxian Qiu

  3. College of Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, China

    Jun Zhu

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  1. Dongmi Luo
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  4. Yibing Chen
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Correspondence to Yibing Chen.

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The research is supported partly by Science Challenge Project (China), No. TZ2016002, National Natural Science Foundation-Joint Fund (China) Grant U1630247, and National Natural Science Foundation of China (Grant Nos. 11671050)

Appendix

Appendix

Fig. 15
figure 15

\(N=200\). Solid line: the reference solution; square symbols and solid line: LLF flux; delta symbols and solid line: NOK flux. Top: \(P^1\) elements; Bottom: \(P^2\) elements

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Fig. 16
figure 16

\(N=5000\). Solid line: the reference solution; square symbols and solid line: LLF flux; delta symbols and solid line: NOK flux. Top: \(P^1\) elements; Bottom: \(P^2\) elements

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In this paper, the NOK flux is employed. But the Lax-Friedrichs flux can be used for our method. The idea is the same as previous section described. However, the formulation is a little different from the present one. In (3.4), the NOK flux is replaced by the local Lax-Friedrichs (LLF) flux, which has the form of

$$\begin{aligned} \hat{F}(a,b)=\frac{1}{2}(F(a)+F(b)-\alpha (b-a)), \end{aligned}$$

where \(\alpha \) is the numerical viscosity constant taken as the largest eigenvalues in magnitude of

$$\begin{aligned} \frac{\partial }{\partial u}F({\bar{u}}_j), \; \frac{\partial }{\partial u}F({\bar{u}}_{j+1}), \end{aligned}$$

where \({\bar{u}}_j\) and \({\bar{u}}_{j+1}\) are the cell averages on the cell \(I_j\) and \(I_{j+1}\), respectively.

Then the same procedure is implemented as Sect. 3.2. The details are omitted here. The final discretization for the volume fraction equation is given by

$$\begin{aligned} \frac{dY_j^{(l)}}{dt}=-&\frac{1}{a_l}[(v_0Y_{j+ \frac{1}{2}}^-+v_0Y_{j+\frac{1}{2}}^+-\alpha (Y_{j+ \frac{1}{2}}^+-Y_{j+\frac{1}{2}}^-))\varphi _l(x_{j+ \frac{1}{2}}^-)-\\&(v_0Y_{j-\frac{1}{2}}^-+v_0Y_{j-\frac{1}{2}}^+- \alpha (Y_{j-\frac{1}{2}}^+-Y_{j-\frac{1}{2}}^-)) \varphi _l(x_{j-\frac{1}{2}}^+))-\\&v_0\int _{I_j}Y(\varphi _l(x))_x dx-Y(x_j)(\hat{v}_{j+ \frac{1}{2}}\varphi _l(x_{j+\frac{1}{2}}^-)-\\&\hat{v}_{j-\frac{1}{2}}\varphi _l(x_{j-\frac{1}{2}}^+) -\int _{I_j}v(\varphi _l(x))_xdx)], \; l=1,\ldots , L, \end{aligned}$$

where \(\hat{v}_{j+\frac{1}{2}}=\frac{1}{2}(v_{j+\frac{1}{2}}^-+v_{j+\frac{1}{2}}^+)\).

Finally, we compute two examples to show the differences of two numerical fluxes and the parameter \(\eta \) in the NOK flux is set to be 0.8. The first one is Shu-Osher problem [25], which contains both shocks and complex smooth region structures. The computational domain is taken as \((-5,5)\). The initial condition is given by

$$\begin{aligned} (\rho ,v,P,\gamma ,B)= {\left\{ \begin{array}{ll} (3.857143,2.629369,\frac{31}{3},1.4,1), \; &{} x\leqslant -4,\\ (1+0.2\sin (5x),0,1,1.9,0), \; &{}x>-4, \end{array}\right. } \end{aligned}$$

The computed density of two numerical fluxes at \(T=1.8\) is plotted in Fig. 15, where the solid line is the fine grid solution computed by \(\Delta x=\frac{1}{2000}\) with \(P^1\) elements.

We recompute the Example 3 using LLF flux. The numerical solution of the density with two numerical fluxes is shown in Fig. 16. From the figures, we can observe that the solution with NOK flux is better than the one with LLF flux for \(P^1\) elements. The differences between them are reduced for \(P^2\) elements.

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Luo, D., Qiu, J., Zhu, J. et al. A Quasi-Conservative Discontinuous Galerkin Method for Multi-component Flows Using the Non-oscillatory Kinetic Flux. J Sci Comput 87, 96 (2021). https://doi.org/10.1007/s10915-021-01494-z

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