Stability of Semi-Lagrangian schemes of arbitrary odd degree under constant and variable advection speed

Ferretti, Roberto; Mehrenberger, Michel

doi:10.1090/mcom/3494

Cited by 8 publications

(7 citation statements)

References 13 publications

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“…Proof Following [21,22], we sketch the arguments to prove (34) for the cases of symmetric Lagrange and splines interpolation. In these cases, the method can be interpreted as Lagrange--Galerkin schemes with area-weighting.…”

Section: Stabilitymentioning

confidence: 99%

“…For a formal definition of the basis functions ψ j in the case of symmetric Lagrange and spline interpolation, we refer the reader to [21,22]. While these two cases allow for a complete theory, at least in one space dimension, in the numerical tests with unstructured grids we will also use P 2 interpolants, for which a first attempt of stability analysis in presented in [23].…”

Section: Stabilitymentioning

confidence: 99%

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Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

et al. 2021

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We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which is based on a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. The proposed method allows to use large time steps, while avoiding the solution of large linear systems, which would be required by implicit time discretization techniques. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. A novel extrapolation technique is proposed to enforce second-order accurate Dirichlet boundary conditions. We include a theoretical analysis of the scheme, along with numerical experiments which demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.

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Section: Stabilitymentioning

confidence: 99%

Section: Stabilitymentioning

confidence: 99%

Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

et al. 2021

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show abstract

“…Note also that, in SL schemes (see, e.g., the discussion of this point in [7]), it is usually required for stability reasons that characteristics passing through neighbouring nodes do not cross. In practice, X ∆ (x i , t n+1 ; t n ) and X ∆ (x k , t n+1 ; t n ) must always have a positive distance, so that, using (4) and the reverse triangular inequality,…”

Section: Element Search By Barycentric Walkmentioning

confidence: 99%

Efficient implementation of characteristic-based schemes on unstructured triangular grids

Cacace¹,

Ferretti²

2021

Preprint

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Using characteristics to treat advection terms in time-dependent PDEs leads to a class of schemes, e.g., semi-Lagrangian and Lagrange-Galerkin schemes, which preserve stability under large Courant numbers, and may therefore be appealing in many practical situations. Unfortunately, the need of locating the feet of characteristics may cause a serious drop of efficiency in the case of unstructured space grids, and thus prevent the use of large time-step schemes on complex geometries.In this paper, we perform an in-depth analysis of the main recipes available for characteristic location, and propose a technique to improve the efficiency of this phase, using additional information related to the advecting vector field. This results in a clear improvement of execution times in the unstructured case, thus extending the range of applicability of large time-step schemes.

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“…The following proposition implies stability for the linear part of the scheme with respect to the 2-norm. For a formal definition of the basis functions ψ j we refer the reader to [17], [18].…”

Section: Stabilitymentioning

confidence: 99%

“…We will therefore focus on the case of high-order interpolations, for which the norm in (30) should be understood as 2-norm. Following [17], [18], we sketch the arguments to prove (30) for the cases of symmetric Lagrange and splines interpolation, which can be interpreted as Lagrange-Galerkin schemes with area-weighting. First, we make explicit the dependence of the points z k on x and ∆t.…”

Section: Stabilitymentioning

confidence: 99%

Second order fully semi-Lagrangian discretizations of advection-diffusion-reaction systems

Bonaventura¹,

Carlini²,

Calzola³

et al. 2020

Preprint

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We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which employs a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. The proposed method allows for large time steps, while avoiding the solution of large linear systems, which would be required by an implicit time discretization technique. Numerical experiments demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.

show abstract

Stability of Semi-Lagrangian schemes of arbitrary odd degree under constant and variable advection speed

Cited by 8 publications

References 13 publications

Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

Efficient implementation of characteristic-based schemes on unstructured triangular grids

Second order fully semi-Lagrangian discretizations of advection-diffusion-reaction systems

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