Recent works have shown that neural networks are promising parameter-free limiters for a variety of numerical schemes (Morgan et al. in A machine learning approach for detecting shocks with high-order hydrodynamic methods. https://doi.org/10.2514/6.2020-2024; Ray et al. in J Comput Phys 367: 166–191. https://doi.org/10.1016/j.jcp.2018.04.029, 2018; Veiga et al. in European Conference on Computational Mechanics and VII European Conference on Computational Fluid Dynamics, vol. 1, pp. 2525–2550. ECCM. https://doi.org/10.5167/uzh-168538, 2018). Following this trend, we train a neural network to serve as a shock-indicator function using simulation data from a Runge-Kutta discontinuous Galerkin (RKDG) method and a modal high-order limiter (Krivodonova in J Comput Phys 226: 879–896. https://doi.org/10.1016/j.jcp.2007.05.011, 2007). With this methodology, we obtain one- and two-dimensional black-box shock-indicators which are then coupled to a standard limiter. Furthermore, we describe a strategy to transfer the shock-indicator to a residual distribution (RD) scheme without the need for a full training cycle and large dataset, by finding a mapping between the solution feature spaces from an RD scheme to an RKDG scheme, both in one- and two-dimensional problems, and on Cartesian and unstructured meshes. We report on the quality of the numerical solutions when using the neural network shock-indicator coupled to a limiter, comparing its performance to traditional limiters, for both RKDG and RD schemes.
In this paper, we present a conservative semi-Lagrangian scheme designed for the numerical solution of 3D hydrostatic free surface flows involving sediment transport on unstructured Voronoi meshes. A high-order reconstruction procedure is employed for obtaining a piecewise polynomial representation of the velocity field and sediment concentration within each control volume. This is subsequently exploited for the numerical integration of the Lagrangian trajectories needed for the discretization of the nonlinear convective and viscous terms. The presented method is fully conservative by construction, since the transported quantity or the vector field is integrated for each cell over the deformed volume obtained at the foot of the characteristics that arises from all the vertexes defining the computational element. The semi-Lagrangian approach allows the numerical scheme to be unconditionally stable for what concerns the advection part of the governing equations. Furthermore, a semi-implicit discretization permits to relax the time step restriction due to the acoustic impedance, hence yielding a stability condition which depends only on the explicit discretization of the viscous terms. A decoupled approach is then employed for the hydrostatic fluid solver and the transport of suspended sediment, which is assumed to be passive. The accuracy and the robustness of the resulting conservative semi-Lagrangian scheme are assessed through a suite of test cases and compared against the analytical solution whenever is known. The new numerical scheme can reach up to fourth order of accuracy on general orthogonal meshes composed by Voronoi polygons.
Hyperbolic systems of conservation laws in multiple spatial dimensions display features absent in the one-dimensional case, such as involutions and non-trivial stationary states. These features need to be captured by numerical methods without excessive grid refinement. The active flux method is an extension of the finite volume scheme with additional point values distributed along the cell boundary. For the equations of linear acoustics, an exact evolution operator can be used for the update of these point values. It incorporates all multi-dimensional information. The active flux method is stationarity preserving, i.e., it discretizes all the stationary states of the PDE. This paper demonstrates the experimental evidence for the discrete stationary states of the active flux method and shows the evolution of setups towards a discrete stationary state.