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A Quasi-Conservative Alternative WENO Finite Difference Scheme for Solving Compressible Multicomponent Flows

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Abstract

We construct a quasi-conservative alternative WENO finite difference scheme respectively coupled with the global Lax-Friedrichs (AWENO-GLF) and the contact restored Harten-Lax-van Leer approximate Riemann solver (AWENO-HLLC) for solving compressible multicomponent flows. The mass equation, the momentum equation, and the energy equation are discretized by a fully conservative AWENO-GLF or AWENO-HLLC finite difference scheme from which a consistent nonconservative discretization of the topological equation is derived according to the velocity and pressure equilibrium principle proposed by Agrall (J Comput Phys 125:150–160, 1996). We prove that, coupling with the constructed scheme, WENO interpolations with common weights for conservative variables or standard WENO interpolations with independent weights for primitive quantities can maintain velocity and pressure equilibrium. Numerical examples demonstrate that AWENO-HLLC scheme is not only less dissipative but also less oscillatory than classical WENO-GLF scheme for compressible multicomponent flows.

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Data availability

The data sets produced in the present study can be obtained from the corresponding author upon reasonable request.

Code availability

The custom codes generated in the course of this study can be obtained from the corresponding author on reasonable request.

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Funding

This work was partially supported by the National Natural Science Foundation of China (Contract Nos. 11901602 and 62231016).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, China

    Yanan Yang & Hua Shen

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100094, China

    Zhiwei He

Authors
  1. Yanan Yang
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  2. Hua Shen
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Correspondence to Hua Shen.

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Appendix: Calculation of common weights

Appendix: Calculation of common weights

The common weights \(\omega _{k}(com)(k=0,1,2)\) are defined as follows [16]:

  1. (1)

    Define the special smooth factor \(\tilde{\beta }(q)\) for the variable q,

    $$\begin{aligned} \begin{aligned} \tilde{\beta }(q)=\frac{\sum _{k=0}^{2}(\beta _{k} (q)+\varepsilon )}{min(\beta _{0}(q), \beta _{1}(q),\beta _{2}(q))+\varepsilon }, \end{aligned} \end{aligned}$$
    (56)

    where \(\beta _k(q)\) (\(k=0,1,2\)) are calculated by (19), and \(\varepsilon \) is an adaptive number defined as

    $$\begin{aligned} \begin{aligned} \varepsilon =\left\{ \begin{array}{ll} \varepsilon _{max} &{} \quad \tau _{5}\le S_{min} \\ \varepsilon _{min} &{} \quad \tau _{5}\ge S_{max} \\ \frac{\varepsilon _{min}-\varepsilon _{max}}{S_{max}-S_{min}} (\tau _{5}-S_{min})+\varepsilon _{max} &{} \quad otherwise, \end{array} \right. \end{aligned} \end{aligned}$$
    (57)

    where \(\varepsilon _{min}=10^{-6}\), \(\varepsilon _{max}=10^{-2}\), \(S_{min}=10^{-3}\), and \(S_{max}=10^{-1}\). \(\tau _{5}=|\beta _{2}(q)-\beta _{0}(q)|\) is the 5th-order global smoothness indicator used in WENO-Z scheme [5].

  2. (2)

    Calculate \(\tilde{\beta }(\alpha ), \tilde{\beta }(\rho )\), and \(\tilde{\beta }(\rho E)\). Then select the variable \(\chi \) from \(\{\alpha ,\rho ,\rho E\}\) with the maximum smooth factor \(\tilde{\beta }\).

  3. (3)

    The common weights are calculated as

    $$\begin{aligned} \begin{aligned} \tilde{a}_{k}=\frac{d_{k}}{(\beta _{k}(\chi )+\varepsilon )^{2}},\ \ \ \omega _{k}(com)=\frac{\tilde{a}_{k}}{\sum _{s=0}^{2}\tilde{a}_{s}}, \end{aligned} \end{aligned}$$
    (58)

    where \(d_{k}\) are optimal linear weights and \(\varepsilon \) is set as shown in (57).

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Yang, Y., Shen, H. & He, Z. A Quasi-Conservative Alternative WENO Finite Difference Scheme for Solving Compressible Multicomponent Flows. J Sci Comput 100, 87 (2024). https://doi.org/10.1007/s10915-024-02645-8

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  • DOI: https://doi.org/10.1007/s10915-024-02645-8

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