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A Flux Reconstruction Stochastic Galerkin Scheme for Hyperbolic Conservation Laws

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Abstract

The study of uncertainty propagation poses a great challenge to design high fidelity numerical methods. Based on the stochastic Galerkin formulation, this paper addresses the idea and implementation of the first flux reconstruction scheme for hyperbolic conservation laws with random inputs. High-order numerical approximation is adopted simultaneously in physical and random space, i.e., the modal representation of solutions is based on an orthogonal polynomial basis and the nodal representation is based on solution collocation points. Therefore, the numerical behaviors of the scheme in the (physical-random) phase space can be designed and understood uniformly. A family of filters is developed in multi-dimensional cases to mitigate the Gibbs phenomenon arising from discontinuities in both physical and random space. The filter function is switched on and off by the dynamic detection of discontinuous solutions, and a slope limiter is employed to preserve the positivity of physically realizable solutions. As a result, the proposed method is able to capture the stochastic flow evolution where resolved and unresolved regions coexist. Numerical experiments including a wave propagation, a Burgers’ shock, a one-dimensional Riemann problem, and a two-dimensional shock-vortex interaction problem are presented to validate the current scheme. The order of convergence of the high-order scheme is identified. The capability of the scheme for simulating smooth and discontinuous stochastic flow dynamics is demonstrated. The open-source codes to reproduce the numerical results are available under the MIT license (Xiao et al. in FRSG: stochastic Galerkin method with flux reconstruction. https://github.com/CSMMLab/FRSG, (2021). https://doi.org/10.5281/zenodo.5588317).

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Data Availability

The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.

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Funding

Tianbai Xiao is funded by the Alexander von Humboldt Foundation (Ref3.5-CHN-1210132-HFST-P). Jonas Kusch is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173. The financial support of the CogniGron research center and the Ubbo Emmius Funds (University of Groningen) is acknowledged.

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Authors and Affiliations

  1. State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, China

    Tianbai Xiao

  2. Steinbuch Centre for Computing, Karlsruhe Institute of Technology, Karlsruhe, Germany

    Martin Frank

  3. Institute of Mathematics, University of Innsbruck, Innsbruck, Austria

    Jonas Kusch

  4. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Groningen, Netherlands

    Julian Koellermeier

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  1. Tianbai Xiao
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  2. Jonas Kusch
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  3. Julian Koellermeier
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Tianbai Xiao, Jonas Kusch and Julian Koellermeier. The first draft of the manuscript was written by Tianbai Xiao and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Appendix Parameter Choice for Exponential Filter

Appendix Parameter Choice for Exponential Filter

While the Lasso filter does not require numerical parameter choices, the exponential filter from Sect. 4.1.1 uses several parameters which need to be determined in applications Table (13).

Table 13 Nomenclature
Full size table

Different strategies exist in the literature. In [26] the filter parameter is chosen as \(\alpha = 36\), together with the filter exponent \(s=36\) to ensure that the last mode is damped to zero up to machine precision. However, the effect on the solution behavior is not clarified. In [31] the parameter choice was motivated with a number of heuristics. Firstly, the effect of the filter on the oscillation of the solution was investigated. Not surprisingly, it was found that larger parameters \(\alpha \) smooth the solution and eventually recover positivity of the filtered distribution function. Secondly, a linear stability analysis of the model linearised around its equilibrium state revealed the damping factors for each mode. It was shown that the choice \(\alpha = 36\) leads to small damping (i.e., less added diffusion) of the solution, while completely damping out the fastest mode. Lastly, the filter was tested with different parameters for the full model and the value \(\alpha = 36\) indeed performed best with respect to the solution quality. While the best choice might depend on the size of the model, the choice of \(\alpha = 36\) was robust in the test cases computed in [31] and this value was therefore used for all further tests computed therein.

In the context of the SG models here, a similar parameter study can be performed to determine a suitable value for the filter parameter. Figure 12 shows the expectation and standard deviation for a simple Burger’s equation test case and different filter parameters \(\alpha \). We choose a constant \(s=3\) as the filter exponent s is only modifying the shape of the filter strength in a mild way. Furthermore, we also choose \(N^* = 0\) fixed as no additional variables need to remain unchanged.

The results in Fig. 12 clearly visualize that a small value of the filter parameter \(\alpha \), e.g., \(\alpha =1\), is not sufficient to damp the oscillations of both the expected values as well as the standard deviation. Similarly, a very large value of the filter parameter, e.g., \(\alpha =60, 100\), also leads to oscillations. In between, there is a range of parameters, for which the oscillations become negligible. This includes the value \(\alpha =36\), which was frequently used in the literature. This indicates that the choice of \(\alpha =36\) also seems to perform well in the settings of this paper and we therefore use it in all test cases including the exponential filter.

Fig. 12
figure 12

Expected value and standard deviation for Burger’s equation and varying filter parameters \(\alpha \) of the exponential filter. The filter exponent is kept fixed at \(s=3\) and we choose \(N_* = 0\)

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Xiao, T., Kusch, J., Koellermeier, J. et al. A Flux Reconstruction Stochastic Galerkin Scheme for Hyperbolic Conservation Laws. J Sci Comput 95, 18 (2023). https://doi.org/10.1007/s10915-023-02143-3

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