Conservative Multimoment Transport along Characteristics for Discontinuous Galerkin Methods

Bosler, Peter Andrew; Bradley, Andrew M.; Taylor, Mark A.

doi:10.1137/18m1165943

Cited by 16 publications

(11 citation statements)

References 51 publications

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“…We name such scheme 'ELDG-CF2' [4], in which 'CF2' refers to the above second order RKEI scheme. When a third order RKEI scheme is used with the ELDG update of linearized solution, one has ) ), ∆t u (1) u (3) = ELDG 2 3 P(u (2) ), ∆t u (1) u n+1 = ELDG − 1 12 P(u (1) ) + 3 4 P(u (3) ), ∆t u (2) .…”

Section: Ordermentioning

confidence: 99%

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An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

Cai,

Qiu,

Yang

2020

Preprint

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We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method. The method is designed as a generalization of the semi-Lagrangian (SL) DG method for linear advection problems proposed in [J. Sci. Comput. 73: 514-542, 2017], which is formulated based on an adjoint problem and tracing upstream cells by tracking characteristics curves highly accurately. In the SLDG method, depending on the velocity field, upstream cells could be of arbitrary shape. Thus, a more sophisticated approximation to sides of the upstream cells is required to get high order approximation. For example, quadratic-curved (QC) quadrilaterals were proposed to approximate upstream cells for a third-order spatial accuracy in a swirling deformation example. In this paper, for linear advection problems, we propose a more general formulation, named the ELDG method. The scheme is formulated based on a modified adjoint problem for which the upstream cells are always quadrilaterals, which avoids the need to use QC quadrilaterals in the SLDG algorithm. The newly proposed ELDG method can be viewed as a new general framework, in which both the classical Eulerian Runge-Kutta DG formulation and the SL DG formulation can fit in. Numerical results on linear transport problems, as well as the nonlinear Vlasov and incompressible Euler dynamics using the exponential RK time integrators, are presented to demonstrate the effectiveness of the ELDG method.

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Section: Ordermentioning

confidence: 99%

“…Compared with ELLAM, the SLDG [5] is being developed in the discontinuous Galerkin finite element framework. SL schemes could be developed base on forward [3] or backward characteristics tracing. Here we choose to develop our schemes base on backward characteristics tracing.…”

Section: Introductionmentioning

confidence: 99%

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

Cai,

Qiu,

Yang

2020

Preprint

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“…QLT is a rich enough algorithm that its implementation could be a topic of another study. The performance study in [7] includes the use of our first implementation of QLT.…”

Section: Level Scheduling In Qltmentioning

confidence: 99%

“…This interpolant can in general cause the method to be unstable, but for p = 2, we have observed that it is empirically linearly stable across a broad range of problems; outside the scope of this paper, we have determined it to be stable analytically for the problem of one-dimensional periodic translation on a uniform mesh. To exercise tracer transport coupled to a different discretization for the total mass continuity equation, we solve for total density using the cell-integrated Jacobian-combined transport method described in [7]. Then the tracer densities must be made consistent with respect to this total density field.…”

Section: Table 72mentioning

confidence: 99%

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Communication-Efficient Property Preservation in Tracer Transport

Bradley¹,

Bosler²,

Guba³

et al. 2019

SIAM J. Sci. Comput.

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Atmospheric tracer transport is a computationally demanding component of the atmospheric dynamical core of weather and climate simulations. Simulations typically have tens to hundreds of tracers. A tracer field is required to preserve several properties, including mass, shape, and tracer consistency. To improve computational efficiency, it is common to apply different spatial and temporal discretizations to the tracer transport equations than to the dynamical equations. Using different discretizations increases the difficulty of preserving properties. This paper provides a unified framework to analyze the property preservation problem and classes of algorithms to solve it. We examine the primary problem and a safety problem; describe three classes of algorithms to solve these; introduce new algorithms in two of these classes; make connections among the algorithms; analyze each algorithm in terms of correctness, bound on its solution magnitude, and its communication efficiency; and study numerical results. A new algorithm, QLT, has the smallest communication volume, and in an important case it redistributes mass approximately locally. These algorithms are only very loosely coupled to the underlying discretizations of the dynamical and tracer transport equations and thus are broadly and efficiently applicable. In addition, they may be applied to remap problems in applications other than tracer transport.

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Petrov–Galerkin flux upwinding for mixed mimetic spectral elements, and its application to geophysical flow problems

Lee

2021

Computers & Mathematics with Applications

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Conservative Multimoment Transport along Characteristics for Discontinuous Galerkin Methods

Cited by 16 publications

References 51 publications

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

Communication-Efficient Property Preservation in Tracer Transport

Petrov–Galerkin flux upwinding for mixed mimetic spectral elements, and its application to geophysical flow problems

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