Abstract
Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques that allow for efficient and accurate simulations while guaranteeing physical properties like mass conservation. In this paper, we develop the first model reduction for the hyperbolic shallow water moment equations and achieve mass conservation. This is accomplished using a macro-micro decomposition of the model into a macroscopic (conservative) part and a microscopic (non-conservative) part with subsequent model reduction using either POD-Galerkin or dynamical low-rank approximation only on the microscopic (non-conservative) part. Numerical experiments showcase the performance of the new model reduction methods including high accuracy and fast computation times together with guaranteed conservation and consistency properties.
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Acknowledgements
The authors were granted access to the HPC resources of IDRIS under the allocation No. AD012A01664R1 attributed by Grand Équipement National de Calcul Intensif (GENCI). Centre de Calcul Intensif d’Aix-Marseille is acknowledged for granting access to its high-performance computing resources. Furthermore, we would like to thank Yannick Schubert for detailed reading and commenting on the work.
Funding
The authors would like to acknowledge the financial support of the CogniGron research center and the Ubbo Emmius Funds (University of Groningen). Jonas Kusch was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 491976834.
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All authors contributed equally to this publication. The original draft and all subsequent edits and reviews were done equally by all authors. The author contributions differ in the following points: Julian Koellermeier: initial idea, methodology, implementation of HSWME, setup of test cases Philipp Krah: implementation of POD-Galerkin, simulation/setup of numerical tests, visualization Jonas Kusch: initial idea, methodology, implementation of HSWME, implementation of DLRA, simulation/setup of numerical tests, visualization
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A Numerical discretization of the decomposed transport part
A Numerical discretization of the decomposed transport part
This appendix briefly describes a numerical scheme for the transport part and then continues with the space-time discretization for the decomposed set of variables Eq. (13).
We first cover a standard scheme. For the spatial discretization, we consider an equidistant mesh \(x_j = j\Delta x \) with lattice spacing \(\Delta x\) and local state-variables, so that the full state vector at time t is given by \(\textbf{Q}(t) = [ {\varvec {q}}_i(x_j,t) ]_{i j} \in \mathbb {R}^{N_x \times (N+2)}\) for \(N_x\) spatial discretization points. Furthermore, we denote the rows of this matrix as \(\textbf{Q}_j(t)\in \mathbb {R}^{N+2}\), which represents the corresponding solution vector evaluated at the spatial cell j.
The spatial discretization of Eq. (2) for each cell j leads to a semi-discrete ODE for the unknowns \(\textbf{Q}_j(t)\). This step is performed via the following standard first-order path-conservative scheme [8, 42]
where the so-called fluctuations \(\textbf{A}^{\pm }_{j+1} \left( \textbf{Q}_{j+1} - \textbf{Q}_{j} \right) \) can be computed in different ways. In this paper, we use the notation from [7] and include the system matrix evaluated at the average of adjacent cells as well as a Lax-Friedrichs type numerical diffusion term for stability as follows
Remark
We note that different numerical schemes are possible that can readily be written in the form of Eq. ()24. Examples are the FORCE scheme
or the upwind scheme
However, these schemes might include higher computational costs. For details, we refer to [41].
Using an explicit Euler time integration scheme for Eq. ()22 yields the time discrete update formula of the transport step
Based on the decomposition of variables and the system matrix given in Eq. () and Eq. (14), a numerical scheme can be formulated.
For the first-order Lax-Friedrichs scheme used in Eq. ()23, the fluctuations for both sets of variables can then be derived as follows.
where we used the abbreviations \(\textbf{A}_{j+1,{\varvec {u}} {\varvec {u}}}:= \textbf{A}_{{\varvec {u}} {\varvec {u}}}\left( \frac{\textbf{U}_j + \textbf{U}_{j+1}}{2},\frac{\textbf{V}_j + \textbf{V}_{j+1}}{2} \right) \) as well as the identity matrices \(\textbf{I}_{2} \in \mathbb {R}^{2 \times 2}\) and \(\textbf{I}_{N} \in \mathbb {R}^{N \times N}\).
The semi-discrete version of Eq. ()22 then reads
The matrices \(\textbf{A}_{j+1,{\varvec {u}} {\varvec {u}}}, \textbf{A}_{j+1, {\varvec {u}} {\varvec {v}}}, \textbf{A}_{j+1, {\varvec {v}} {\varvec {u}}}, \textbf{A}_{j+1, {\varvec {v}} {\varvec {v}}}\) are given by Eq. (14) evaluated at the averaged variables as follows:
where the matrix \(\textbf{A}\in \mathbb {R}^{N\times N}\) is zero except for the off-diagonal entries
The semi-discrete transport step using an explicit Euler time integration scheme for Eq. ()25 leads to the time discrete transport update for the first two equations (macro transport step)
and to the time discrete transport update for the last N moment equations (micro transport step)
Note that the solution of the micro step for \(\textbf{V}_j^{n+1}\) is later performed with the known values \(\widetilde{\textbf{U}}_j^{n+1}\) from the first macro step (see Eq. (5)). The solution of the macro step should be performed with a conservative scheme, to ensure mass conservation.
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Koellermeier, J., Krah, P. & Kusch, J. Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: a study using POD-Galerkin and dynamical low-rank approximation. Adv Comput Math 50, 76 (2024). https://doi.org/10.1007/s10444-024-10175-y
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DOI: https://doi.org/10.1007/s10444-024-10175-y