Academic literature on the topic 'Semi-Lagrangian discretizations'

AbstractWe develop numerical methods for the simulation of laden-flows where particles interact with the carrier fluid through drag forces. Semi-Lagrangian techniques are presented to handle the Vlasov-type equation which governs the evolution of the particles. We discuss several options to treat the coupling with the hydrodynamic system describing the fluid phase, paying attention to strategies based on staggered discretizations of the fluid velocity.

Abstract The discontinuous Galerkin (DG) methods designed for hyperbolic problems arising from a wide range of applications are known to enjoy many computational advantages. DG methods coupled with strong-stability-preserving explicit Runge–Kutta discontinuous Galerkin (RKDG) time discretizations provide a robust numerical approach suitable for geoscience applications including atmospheric modeling. However, a major drawback of the RKDG method is its stringent Courant–Friedrichs–Lewy (CFL) stability restriction associated with explicit time stepping. To address this issue, the authors adopt a dimension-splitting approach where a semi-Lagrangian (SL) time-stepping strategy is combined with the DG method. The resulting SLDG scheme employs a sequence of 1D operations for solving multidimensional transport equations. The SLDG scheme is inherently conservative and has the option to incorporate a local positivity-preserving filter for tracers. A novel feature of the SLDG algorithm is that it can be used for multitracer transport for global models employing spectral-element grids, without using an additional finite-volume grid system. The quality of the proposed method is demonstrated via benchmark tests on Cartesian and cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates.

Abstract A new semi-Lagrangian advection scheme called multistep ray advection is proposed for solving the spectral energy balance equation of ocean surface gravity waves. Existing so-called piecewise ray methods advect wave energy over a single time step using “pieces” of ray trajectories, after which the spectrum is updated with source terms representing various physical processes. The generalized scheme presented here allows for an arbitrary number N of advection time steps along the same rays, thus reducing numerical diffusion, and still including source-term variations every time step. Tests are performed for alongshore uniform bottom topography, and the effects of two types of discretizations of the wave spectrum are investigated, a finite-bandwidth representation and a single frequency and direction per spectral band. In the limit of large N, both the accuracy and computation cost of the method increase, approaching a nondiffusive fully Lagrangian scheme. Even for N = 1 the semi-Lagrangian scheme test results show less numerical diffusion than predictions of the commonly used first-order upwind finite-difference scheme. Application to the refraction and shoaling of narrow swell spectra across a continental shelf illustrates the importance of controlling numerical diffusion. Numerical errors in a single-step (Δt = 600 s) scheme implemented on the North Carolina continental shelf (typical swell propagation time across the shelf is about 3 h) are shown to be comparable to the angular diffusion predicted by the wave–bottom Bragg scattering theory, in particular for narrow directional spectra, suggesting that the true directional spread of swell may not always be resolved in existing wave prediction models, because of excessive numerical diffusion. This diffusion is effectively suppressed in cases presented here with a four-step semi-Lagrangian scheme, using the same value of Δt.

We present a novel up-resing technique for generating high-resolution liquids based on scene flow estimation using deep neural networks. Our approach infers and synthesizes small- and large-scale details solely from a low-resolution particle-based liquid simulation. The proposed network leverages neighborhood contributions to encode inherent liquid properties throughout convolutions. We also propose a particle-based approach to interpolate between liquids generated from varying simulation discretizations using a state-of-the-art bidirectional optical flow solver method for fluids in addition with a novel key-event topological alignment constraint. In conjunction with the neighborhood contributions, our loss formulation allows the inference model throughout epochs to reward important differences in regard to significant gaps in simulation discretizations. Even when applied in an untested simulation setup, our approach is able to generate plausible high-resolution details. Using this interpolation approach and the predicted displacements, our approach combines the input liquid properties with the predicted motion to infer semi-Lagrangian advection. We furthermore showcase how the proposed interpolation approach can facilitate generating large simulation datasets with a subset of initial condition parameters.

Abstract A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and stability analysis is proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection diffusion and nonlinear parabolic problems.

Abstract. The semi-Lagrangian absolute vorticity (SL-AV) atmospheric model is the global semi-Lagrangian hydrostatic model used for operational medium-range and seasonal forecasts at the Hydrometeorological Centre of Russia. The distinct feature of the SL-AV dynamical core is the semi-implicit, semi-Lagrangian vorticity-divergence formulation on the unstaggered grid. A semi-implicit, semi-Lagrangian approach allows for long time steps but violates the global and local mass conservation. In particular, the total mass in simulations with semi-Lagrangian models can drift significantly if no a posteriori mass-fixing algorithm is applied. However, the global mass-fixing algorithms degrade the local mass conservation. The new inherently mass-conservative version of the SL-AV model dynamical core presented here ensures global and local mass conservation without mass-fixing algorithms. The mass conservation is achieved with the introduction of the finite-volume, semi-Lagrangian discretization for a continuity equation based on the 3-D extension of the conservative cascade semi-Lagrangian transport scheme (CCS). Numerical experiments show that the new version of the SL-AV dynamical core presented combines the accuracy and stability of the standard SL-AV dynamical core with the mass-conservation properties. The results of the mountain-induced Rossby-wave test and baroclinic instability test for the mass-conservative dynamical core are found to be in agreement with the results available in the literature.

A esfera é utilizada como domínio computacional na modelagem de diversos fenômenos físicos, como em previsão numérica do tempo. Sua discretização pode ser feita de diversas formas, sendo comum o uso de malha regulares em latitude/longitude. Recentemente, também para melhor uso de computação paralela, há uma tendência ao uso de malhas mais isotrópicas, dentre as quais a icosaédrica. Apesar de já existirem modelos atmosféricos que usam malhas icosaédricas, não há consenso sobre as metodologias mais adequadas a esse tipo de malha. Nos propusemos, portanto, a estudar em detalhe diversos fatores envolvidos no desenvolvimento de modelos atmosféricos globais usando malhas geodésicas icosaédricas. A discretização usual por volumes finitos para divergente de um campo vetorial utiliza como base o Teorema da Divergência e a regra do ponto médio nas arestas das células computacionais. A distribuição do erro obtida com esse método apresenta uma forte relação com características geométricas da malha. Definimos o conceito geométrico de alinhamento de células computacionais e desenvolvemos uma teoria que serve de base para explicar interferências de malha na discretização usual do divergente. Destacamos os impactos de certas relações de alinhamento das células na ordem da discretização do método. A teoria desenvolvida se aplica a qualquer malha geodésica e também pode ser usada para os operadores rotacional e laplaciano. Investigamos diversos métodos de interpolação na esfera adequados a malhas icosaédricas, e abordamos o problema de interpolação e reconstrução vetorial na esfera em malhas deslocadas. Usamos métodos alternativos de reconstrução vetorial aos usados na literatura, em particular, desenvolvemos um método híbrido de baixo custo e boa precisão. Por fim, utilizamos as técnicas de discretização, interpolação e reconstrução vetorial analisadas em um método semi-lagrangiano para o transporte na esfera em malhas geodésicas icosaédricas. Realizamos experimentos computacionais de transporte, incluindo testes de deformações na distribuição do campo transportado, que mostraram a adequação da metodologia para uso em modelos atmosféricos globais. A plataforma computacional desenvolvida nesta tese, incluindo geração de malhas, interpolações, reconstruções vetoriais e um modelo de transporte, fornece uma base para o futuro desenvolvimento de um modelo atmosférico global em malhas icosaédricas.
Spherical domains are used to model many physical phenomena, as, for instance, global numerical weather prediction. The sphere can be discretized in several ways, as for example a regular latitude-longitude grid. Recently, also motivated by a better use of parallel computers, more isotropic grids have been adopted in atmospheric global circulation models. Among those, the icosahedral grids are promising. Which kind of discretization methods and interpolation schemes are the best to use on those grids are still a research subject. Discretization of the sphere may be done in many ways and, recently, to make better use of computational resources, researchers are adopting more isotropic grids, such as the icosahedral one. In this thesis, we investigate in detail the numerical methodology to be used in atmospheric models on icosahedral grids. The usual finite volume method of discretization of the divergence of a vector field is based on the divergence theorem and makes use of the midpoint rule for integration on the edges of computational cells. The error distribution obtained with this method usually presents a strong correlation with some characteristics of the icosahedral grid. We introduced the concept of cell alignment and developed a theory which explains the grid imprinting patterns observed with the usual divergence discretization. We show how grid alignment impacts in the order of the divergence discretization. The theory developed applies to any geodesic grid and can also be used for other operators such as curl and Laplacian. Several interpolation schemes suitable for icosahedral grids were analysed, including the vector interpolation and reconstruction problem on staggered grids. We considered alternative vector reconstruction methods, in particular, we developed a hybrid low cost and good precision method. Finally, employing the discretizations and interpolations previously analysed, we developed a semi-Lagrangian transport method for geodesic icosahedral grids. Several tests were carried out, including deformational test cases, which demonstrated that the methodology is suitable to use in global atmospheric models. The computational platform developed in this thesis, including mesh generation, interpolation, vector reconstruction and the transport model, provides a basis for future development of global atmospheric models on icosahedral grids.