Darren Engwirda

Darren Engwirda, Maxwell Kelley

A new, high-order slope-limiting procedure for the Piecewise Parabolic Method (PPM) and the Piecewise Quartic Method (PQM) is described. Following a Weighted Essentially Non-Oscillatory (WENO)-type paradigm, the proposed slope-limiter seeks to reconstruct smooth, non-oscillatory piecewise polynomial profiles as a non-linear combination of the natural and monotone-limited PPM and PQM interpolants. Compared to existing monotone slope-limiting techniques, this new strategy is designed to improve accuracy at smooth extrema, while controlling spurious oscillations in the neighbourhood of sharp features. Using the new slope-limited PPM and PQM interpolants, a high-order accurate Arbitrary-Lagrangian-Eulerian framework for advection-dominated flows is constructed, and its effectiveness is examined using a series of one- and two-dimensional benchmark cases. It is shown that the new WENO-type slope-limiting techniques offer a significant improvement in accuracy compared to existing strategies, allowing the PPM- and PQM- based schemes to achieve fully third- and fifth-order accurate convergence, respectively, for sufficiently smooth problems.

Darren Engwirda

The generation of high-quality staggered unstructured grids is considered, leading to the development of a new optimisation-based strategy designed to construct weighted `Regular-Power' tessellations appropriate for co-volume type numerical discretisation techniques. This new framework aims to extend the conventional Delaunay-Voronoi primal-dual structure; seeking to assemble generalised orthogonal tessellations with enhanced geometric quality. The construction of these grids is motivated by the desire to improve the performance and accuracy of numerical methods based on unstructured co-volume type schemes, including various staggered grid techniques for the simulation of fluid dynamics and hyperbolic transport. In this study, a new hybrid optimisation strategy is proposed; seeking to optimise the geometry, topology and weights associated with general, two-dimensional Regular-Power tessellations using a combination of gradient-ascent and energy-based techniques. The performance of this new method is tested experimentally, with a range of complex, multi-resolution primal-dual grids generated for various coastal and regional ocean modelling applications.

Darren Engwirda

An algorithm for the generation of non-uniform unstructured grids on ellipsoidal geometries is described. This technique is designed to generate high quality triangular and polygonal meshes appropriate for general circulation modelling on the sphere, including applications to atmospheric and ocean simulation, and numerical weather predication. Using a recently developed Frontal-Delaunay-refinement technique, a method for the construction of high-quality unstructured ellipsoidal Delaunay triangulations is introduced. A dual polygonal grid, derived from the associated Voronoi diagram, is also optionally generated as a by-product. Compared to existing techniques, it is shown that the Frontal-Delaunay approach typically produces grids with near-optimal element quality and smooth grading characteristics, while imposing relatively low computational expense. Initial results are presented for a selection of uniform and non-uniform ellipsoidal grids appropriate for large-scale geophysical applications. The use of user-defined mesh-sizing functions to generate smoothly graded, non-uniform grids is discussed.

Shane X. Coffing, John Tipton, Arvind T. Mohan et al.

Reduced order models (ROM) can represent spatiotemporal processes in significantly fewer dimensions and can be solved many orders faster than their governing partial differential equations (PDEs). For example, using a proper orthogonal decomposition produces a ROM that is a small linear combination of fixed features and weights, but that is constrained to the given process it models. In this work, we explore a new type of ROM that is not constrained to fixed weights, based on neural Galerkin-Projections, which is an initial value problem that encodes the physics of the governing PDEs, calibrated via neural networks to accurately model the trajectory of these weights. Then using a statistical hierarchical pooling technique to learn a distribution on the initial values of the temporal weights, we can create new, statistically interpretable and physically justified weights that are generalized to many similar problems. When recombined with the spatial features, we form a complete physics surrogate, called a randPROM, for generating simulations that are consistent in distribution to a neighborhood of initial conditions close to those used to construct the ROM. We apply the randPROM technique to the study of tsunamis, which are unpredictable, catastrophic, and highly-detailed non-linear problems, modeling both a synthetic case of tsunamis near Fiji and the real-world Tohoku 2011 disaster. We demonstrate that randPROMs may enable us to significantly reduce the number of simulations needed to generate a statistically calibrated and physically defensible prediction model for arrival time and height of tsunami waves.

Edward McDugald, Arvind Mohan, Darren Engwirda et al.

We investigate the potential of an attention-based neural network architecture, the Senseiver, for sparse sensing in tsunami forecasting. Specifically, we focus on the Tsunami Data Assimilation Method, which generates forecasts from tsunameter networks. Our model is used to reconstruct high-resolution tsunami wavefields from extremely sparse observations, including cases where the tsunami epicenters are not represented in the training set. Furthermore, we demonstrate that our approach significantly outperforms the Linear Interpolation with Huygens-Fresnel Principle in generating dense observation networks, achieving markedly improved accuracy.