Semi‐Lagrangian advection on a spherical geodesic grid

Carfora, Maria Francesca

doi:10.1002/fld.1445

Cited by 5 publications

(3 citation statements)

References 29 publications

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“…[19][20][21]), or other regular surface-based grids and local surface-based coordinate systems (e.g. [22][23][24]), which can lead to a loss of accuracy because of singularities that arise in the mappings from the physical sphere to the surface-based coordinate systems.…”

Section: Introductionmentioning

confidence: 99%

Mesh-free semi-Lagrangian methods for transport on a sphere using radial basis functions

Shankar

Wright

2018

Journal of Computational Physics

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We present three new semi-Lagrangian methods based on radial basis function (RBF) interpolation for numerically simulating transport on a sphere. The methods are mesh-free and are formulated entirely in Cartesian coordinates, thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface-based coordinate systems. For problems involving tracer transport in a given velocity field, the semi-Lagrangian framework allows these new methods to avoid the use of any stabilization terms (such as hyperviscosity) during timeintegration, thus reducing the number of parameters that have to be tuned. The three new methods are based on interpolation using 1) global RBFs, 2) local RBF stencils, and 3) RBF partition of unity. For the latter two of these methods, we find that it is crucial to include some low degree spherical harmonics in the interpolants. Standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy, efficiency, conservation properties, and dissipation/dispersion errors. For global RBFs, spectral spatial convergence is observed for smooth solutions on quasi-uniform nodes, while high-order accuracy is observed for the local RBF stencil and partition of unity approaches.

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Section: Introductionmentioning

confidence: 99%

Mesh-free semi-Lagrangian methods for transport on a sphere using radial basis functions

Shankar

Wright

2018

Journal of Computational Physics

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“…Giraldo [20] proved its simplification to the midpoint integration rule on the surface of a sphere. Additionally, the midpoint method was used widespread for the approximation of the characteristic equation in the semi-Lagrangian method [21][22][23]. It is implemented mainly as follows:…”

Section: Ritchie's Methodsmentioning

confidence: 99%

High-Order Semi-Lagrangian Schemes for the Transport Equation on Icosahedron Spherical Grids

Zhang

Tian

et al. 2022

Atmosphere

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The transport process is an important part of the research of fluid dynamics, especially when it comes to tracer advection in the atmosphere or ocean dynamics. In this paper, a series of high-order semi-Lagrangian methods for the transport process on the sphere are considered. The methods are formulated entirely in three-dimensional Cartesian coordinates, thus avoiding any apparent artificial singularities associated with surface-based coordinate systems. The underlying idea of the semi-Lagrangian method is to find the value of the field/tracer at the departure point through interpolating the values of its surrounding grid points to the departure point. The implementation of the semi-Lagrangian method is divided into the following two main procedures: finding the departure point by integrating the characteristic equation backward and then interpolate on the departure point. In the first procedure, three methods are utilized to solve the characteristic equation for the locations of departure points, including the commonly used midpoint-rule method and two explicit high-order Runge–Kutta (RK) methods. In the second one, for interpolation, four new methods are presented, including 1) linear interpolation; 2) polynomial fitting based on the least square method; 3) global radial basis function stencils (RBFs), and 4) local RBFs. For the latter two interpolation methods, we find that it is crucial to select an optimal value for the shape parameter of the basis function. A Gauss hill advection case is used to compare and contrast the methods in terms of their accuracy, and conservation properties. In addition, the proposed method is applied to standard test cases, which include solid body rotation, shear deformation of twin slotted cylinders, and the evolution of a moving vortex. It demonstrates that the proposed method could simulate all test cases with reasonable accuracy and efficiency.

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“…Nota-se ainda que há um investimento crescente de alguns grupos em possíveis melhorias para os métodos de integração no tempo (Lipscomb e , Miura [2007], Skamarock e Menchaca [2010a]), geralmente no sentido de obter um método monotônico e conservativo. Em Carfora [2007b] há um estudo simplificado a respeito de um método semi-lagrangiano para malha icosaédricas.…”

Section: Breve Históricounclassified

Análise de discretizações e interpolações em malhas icosaédricas e aplicações em modelos de transporte semi-lagrangianos

Peixoto¹

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Uma tese de doutorado é muito mais que apenas um conjunto de páginas escritas. A tese é na verdade um conjunto de experiências vividas. São as experiências que fazem o amadurecimento científico acontecer, e isso geralmente não está explícito na tese. Esta tese foi realizada num período de 4 anos, nos quais muitas coisas aconteceram e impactaram nos resultados que serão apresentados. Agradecimentos Muitas pessoas contribuíram para o desenvolvimento deste trabalho de doutorado, mas sem dúvidas, o meu orientador, Saulo, é quem mais contribuiu, com muitas horas discussões e ensinamentos. Tenho que agradecer também os professores do IME que se dispuseram a me ensinar matemática de alto nível. Agradeço também os membros da banca de qualificação, Pedro Dias e Abimael Loula, que deram sugestões que impulsionaram o trabalho e incentivaram a publicação dos resultados que havia obtido até então. A base de trabalho foi o laboratório de matemática aplicada, LabMap, que não teria a qualidade que tem hoje sem o esforço do professor Alexandre Roma. Sempre que tinha um problema colocava na lousa do laboratório e discutia com alguém e daí diversas ideias surgiam, portanto tem um pouco de cada amigo do IME nesta tese. Aí vão alguns nomes importantes que merecem um obrigado:

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Semi‐Lagrangian advection on a spherical geodesic grid

Cited by 5 publications

References 29 publications

Mesh-free semi-Lagrangian methods for transport on a sphere using radial basis functions

Mesh-free semi-Lagrangian methods for transport on a sphere using radial basis functions

High-Order Semi-Lagrangian Schemes for the Transport Equation on Icosahedron Spherical Grids

Análise de discretizações e interpolações em malhas icosaédricas e aplicações em modelos de transporte semi-lagrangianos

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