Exact Projections and the Lagrange-Galerkin Method: A Realistic Alternative to Quadrature

Priestley, A.

doi:10.1006/jcph.1994.1104

Cited by 43 publications

(34 citation statements)

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“…functions evaluated at the departure points of the trajectories arriving at the quadrature nodes, cannot be computed exactly (see e.g. [28,34]), since such functions are not polynomials. Therefore, a sufficiently accurate approximation of these integrals is needed, which may entail the need to employ numerical quadrature formulae with more nodes than the minimal requirement implied by the local polynomial degree.…”

Section: Substituting Now Expressionsmentioning

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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Section: Substituting Now Expressionsmentioning

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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“…The location of points inside the elements of a mesh is a trivial task in structured meshes, for instance, in meshes composed of squares or hexahedra, but if the mesh is unstructured the location of points is not that simple; hence, LG methods may become less efficient than they look at first. To partially overcome these drawbacks, some variations of conventional LG method, such as the area-weighting method for quadrilateral structured meshes [20], exact integration [22] for straight side triangular meshes with linear elements, and the modified LG methods [5,6], have been proposed. We do not consider such variations in this paper.…”

Section: Introductionmentioning

confidence: 99%

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Bermejo

Saavedra

2016

Communications in Applied and Industrial Mathematics

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We review in this paper the development of Lagrange-Galerkin (LG) methods to integrate the incompressible Navier-Stokes equations (NSEs) for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the flow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers.

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“…In this field, several competitive methods have been developed, in particular one could mention the Eulerian-Lagrangian Localized Adjoint Method (ELLAM), see, e.g., [4], [17], [22], the characteristic Galerkin method ( [31]), the Lagrange-Galerkin method ( [43], [46]) as well as semi-Lagrangian ( [28], [45]) and Free-Lagrangian methods ( [13], [19]). While operating on the basis of the Lagrangian concept, this kind of schemes keeps the advantage of the presentation of the solution on a fixed Eulerian grid.…”

Section: Introductionmentioning

confidence: 99%

A fast high‐resolution algorithm for linear convection problems: particle transport method

Smolianski

Shipilova

Haario

2006

Numerical Meth Engineering

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The paper is devoted to a novel explicit technique, the particle transport method (PTM), for solving linear convection problems. While being a Lagrangian (characteristic based) method, PTM has the advantage of Eulerian methods to represent the solution on a fixed mesh. The proposed approach belongs to the class of monotone high-resolution numerical schemes, possesses the property of unconditional stability and works with structured and unstructured meshes. It is also demonstrated that the method has a linear computational complexity. The performance of the presented algorithm is tested on one-and two-dimensional benchmark problems. The numerical results confirm that the method has the 2nd-order spatial accuracy and can be significantly faster than the grid-based methods of the same order. AbstractThe paper is devoted to a novel explicit technique, the Particle Transport Method (PTM), for solving linear convection problems. While being a Lagrangian (characteristic based) method, PTM has the advantage of Eulerian methods to represent the solution on a fixed mesh. The proposed approach belongs to the class of monotone high-resolution numerical schemes, possesses the property of unconditional stability and works with structured and unstructured meshes. It is also demonstrated that the method has a linear computational complexity. The performance of the presented algorithm is tested on one-and two-dimensional benchmark problems. The numerical results confirm that the method has the 2nd-order spatial accuracy and can be significantly faster than the grid-based methods of the same order.

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Exact Projections and the Lagrange-Galerkin Method: A Realistic Alternative to Quadrature

Cited by 43 publications

References 0 publications

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

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

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

A fast high‐resolution algorithm for linear convection problems: particle transport method

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