A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows

Restelli, Marco; Bonaventura, Luca; Sacco, Riccardo

doi:10.1016/j.jcp.2005.11.030

Cited by 69 publications

(75 citation statements)

References 51 publications

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“…In the future, we plan to implement the pseudo-Helmholtz form of the linear system for the case of general boundary conditions, develop high-order semi-implicit methods with adaptive time-stepping, explore alternative high-order non-reflective boundary conditions, and extend the model to three dimensions to investigate the effects of rotation. A further possible extension is represented by the introduction of the semi-Lagrangian DG method proposed in [41] to deal with the stability limit associated with advection.…”

Section: Discussionmentioning

confidence: 99%

A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier–Stokes Equations in Nonhydrostatic Mesoscale Modeling

Restelli¹,

Giraldo²

2009

SIAM J. Sci. Comput.

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Abstract.A discontinuous Galerkin (DG) finite element formulation is proposed for the solution of the compressible Navier-Stokes equations for a vertically stratified fluid, which are of interest in mesoscale nonhydrostatic atmospheric modeling. The resulting scheme naturally ensures conservation of mass, momentum, and energy. A semi-implicit time-integration approach is adopted to improve the efficiency of the scheme with respect to the explicit Runge-Kutta time integration strategies usually employed in the context of DG formulations. A method is also presented to reformulate the resulting linear system as a pseudo-Helmholtz problem. In doing this, we obtain a DG discretization closely related to those proposed for the solution of elliptic problems, and we show how to take advantage of the numerical integration rules (required in all DG methods for the area and flux integrals) to increase the efficiency of the solution algorithm. The resulting numerical formulation is then validated on a collection of classical two-dimensional test cases, including density driven flows and mountain wave simulations. The performance analysis shows that the semi-implicit method is, indeed, superior to explicit methods and that the pseudo-Helmholtz formulation yields further efficiency improvements. 1. Introduction. In recent years, great attention has been devoted to the discontinuous Galerkin (DG) finite element method in the context of geophysical fluid dynamics applications. This is motivated by the fact that the DG framework simultaneously provides a high-order discretization, great flexibility in the choice of the computational grid, discrete balance relations, robustness with respect to unphysical oscillations, and compact computational stencils which are a key element in order to exploit distributed-memory parallel computers with up to tens of thousands of processors. Without attempting to provide a complete review of the literature, we mention here [47,2,30,39,34,32], where DG shallow water models are presented. The application of the DG method to compressible, nonhydrostatic atmospheric flows, using the Navier-Stokes equations or, when the flow is assumed to be inviscid, the Euler equations, is then considered in [31], where it is shown that the method represents a good candidate for the development of numerical climate and weather models. In the present paper, we continue the study initiated in [31] by focusing on the aspect of the time discretization which is, in fact, the most penalizing drawback of the DG method due to its high computational cost. This latter cost stems from the following

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Section: Discussionmentioning

confidence: 99%

A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier–Stokes Equations in Nonhydrostatic Mesoscale Modeling

Restelli¹,

Giraldo²

2009

SIAM J. Sci. Comput.

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“…the Richards equation, and to the extension of this technique to high order discontinuous finite elements discretizations such as those proposed in [15], [35], [36]. …”

Section: Discussionmentioning

confidence: 99%

“…Some of the earliest and best known techniques of this kind were proposed in a finite volume framework in [10], [11] for applications to numerical weather prediction and climate modelling. Similar ideas have also been proposed in [12], [13], [14], for applications to plasma modelling and flow in porous media, while extensions of this ideas to the Discontinuous Galerkin framework have been first proposed in [15] and later also in [16], [17], [18]. All these methods are exactly mass conservative, as opposed to methods based on upstream remapping of computational cells (also called inherently conservative in the numerical weather prediction literature).…”

Section: Introductionmentioning

confidence: 98%

Flux form Semi-Lagrangian methods for parabolic problems

Bonaventura

Ferretti

2016

Communications in Applied and Industrial Mathematics

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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.

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“…The conservation of mass itself is attained here by using a mixed Eulerian (for the continuity equation in flux form) semi-Lagrangian (for the momentum equation) formulation, and a more ambitious extension to a fully conservative and semi-Lagrangian formulation on the lines of [11] is left for further studies.…”

Section: Conclusion and Future Perspectivesmentioning

confidence: 99%

“…It has to be stressed that both of these extensions are only a first attempt to generalize the numerical formulation of [1], while a more ambitious extension to a fully conservative and semi-Lagrangian approach on the lines of [11], as well as an implementation on generical nonstructured meshes, is left for further studies.…”

Section: Introductionmentioning

confidence: 99%

A mass conservative TR-BDF2 semi-implicit semi-Lagrangian DG discretization of the shallow water equations on general structured meshes of quadrilaterals

Tumolo

2016

Communications in Applied and Industrial Mathematics

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As an extension of a previous work considering a fully advective formulation on Cartesian meshes, a mass conservative discretization approach is presented here for the shallow water equations, based on discontinuous finite elements on general structured meshes of quadrilaterals. A semi-implicit time integration is performed by employing the TR-BDF2 scheme and is combined with the semi-Lagrangian technique for the momentum equation only. Indeed, in order to simplify the derivation of the discrete linear Helmoltz equation to be solved at each time-step, a non-conservative formulation of the momentum equation is employed. The Eulerian flux form is considered instead for the continuity equation in order to ensure mass conservation. Numerical results show that on distorted meshes and for relatively high polynomial degrees, the proposed numerical method fully conserves mass and presents a higher level of accuracy than a standard off-centered Crank Nicolson approach. This is achieved without any significant imprinting of the mesh distortion on the solution.

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A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows

Cited by 69 publications

References 51 publications

A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier–Stokes Equations in Nonhydrostatic Mesoscale Modeling

A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier–Stokes Equations in Nonhydrostatic Mesoscale Modeling

Flux form Semi-Lagrangian methods for parabolic problems

A mass conservative TR-BDF2 semi-implicit semi-Lagrangian DG discretization of the shallow water equations on general structured meshes of quadrilaterals

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