In this work, a novel artificial viscosity method is proposed using smooth and compactly supported viscosities. These are derived by revisiting the widely used piecewise constant artificial viscosity method of Persson and Peraire as well as the piecewise linear refinement of Klöckner et al. with respect to the fundamental design criteria of conservation and entropy stability. Further investigating the method of modal filtering in the process, it is demonstrated that this strategy has inherent shortcomings, which are related to problems of Legendre viscosities to handle shocks near element boundaries. This problem is overcome by introducing certain functions from the fields of robust reprojection and mollififers as viscosity distributions. To the best of our knowledge, this is proposed for the first time in this work. The resulting C ∞ 0 artificial viscosity method is demonstrated to provide sharper profiles, steeper gradients and a higher resolution of small-scale features while still maintaining stability of the method.MSC2010: 35L65, 65N30, 65N35, 35L67, 35D40

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General circulation models are essential tools in weather and hydrodynamic simulation. They solve discretized, complex physical equations in order to compute evolutionary states of dynamical systems, such as the hydrodynamics of a lake. However, high-resolution numerical solutions using such models are extremely computational and time consuming, often requiring a high performance computing architecture to be executed satisfactorily. Machine learning (ML)-based low-dimensional surrogate models are a promising alternative to speed up these simulations without undermining the quality of predictions. In this work, we develop two examples of fast, reliable, low-dimensional surrogate models to produce a 36 hour forecast of the depth-averaged hydrodynamics at Lake George NY, USA. Our ML approach uses two widespread artificial neural network (ANN) architectures: fully connected neural networks and long short-term memory. These ANN architectures are first validated in the deterministic and chaotic regimes of the Lorenz system and then combined with proper orthogonal decomposition (to reduce the dimensionality of the incoming input data) to emulate the depth-averaged hydrodynamics of a flow simulator called SUNTANS. Results show the ANN-based reduced order models have promising accuracy levels (within 6% of the prediction range) and advocate for further investigation into hydrodynamic applications.

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