An Adaptive Algorithm for the Time Dependent Transport Equation with Anisotropic Finite Elements and the Crank–Nicolson Scheme

Dubuis, Samuel; Picasso, Marco

doi:10.1007/s10915-017-0537-1

Cited by 6 publications

(13 citation statements)

References 19 publications

(45 reference statements)

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“…To approximate in space (1) a finite element stabilization scheme is needed. This scheme corresponds to the stabilized scheme studied in [12], in which the stabilization term is updated to the anisotropic setting, following [11,21,29].…”

Section: Statement Of the Problem And Numerical Schemementioning

confidence: 99%

“…Moreover, standard a posteriori proofs lead to suboptimal estimates for second order methods, as reported in [2]. To circumvent this problem, piecewise quadratic reconstructions have been introduced for parabolic problems [1,2,7,28], in [21] for the transport equations and in [27] for the wave equation.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In this paper, the a posteriori error analysis of [21] for the transport equation is extended to the case where the velocity varies in space and time. As in [21], the a posteriori error analysis is valid for anisotropic meshes and a second order time discretization. In [21], the a posteriori error estimate was restricted to the case of a steady velocity while the new estimate is valid for velocity fields that may exhibit strong variations in time.…”

Section: Introductionmentioning

confidence: 99%

“…As in [21], the a posteriori error analysis is valid for anisotropic meshes and a second order time discretization. In [21], the a posteriori error estimate was restricted to the case of a steady velocity while the new estimate is valid for velocity fields that may exhibit strong variations in time. The techniques used should be useful when coupling the transport equation to Navier-Stokes equations, which is the case for instance when considering free surface flows [20].…”

Section: Introductionmentioning

confidence: 99%

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An adaptive algorithm for the transport equation with time dependent velocity

Dubuis

Picasso

2020

SN Appl. Sci.

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An a posteriori error estimate is derived for the approximation of the transport equation with a time dependent transport velocity. Continuous, piecewise linear, anisotropic finite elements are used for space discretization, the Crank-Nicolson scheme scheme is proposed for time discretization. This paper is a generalization of Dubuis S, Picasso M (J Sci Comput 75(1):350–375, 2018) where the transport velocity was not depending on time. The a posteriori error estimate (upper bound) is shown to be sharp for anisotropic meshes, the involved constant being independent of the mesh aspect ratio. A quadratic reconstruction of the numerical solution is introduced in order to obtain an estimate that is order two in time. Error indicators corresponding to space and time are proposed, their accuracy is checked with non-adapted meshes and constant time steps. Then, an adaptive algorithm is introduced, allowing to adapt the meshes and time steps. Numerical experiments are presented when the exact solution has strong variations in space and time, illustrating the efficiency of the method. They indicate that the effectivity index is close to one and does not depend on the solution, mesh size, aspect ratio, and time step.

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Section: Statement Of the Problem And Numerical Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An adaptive algorithm for the transport equation with time dependent velocity

Dubuis

Picasso

2020

SN Appl. Sci.

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Time‐continuous and time‐discontinuous space‐time finite elements for advection‐diffusion problems

Danwitz

Voulis

Hosters

et al. 2023

Numerical Meth Engineering

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We construct four variants of space‐time finite element discretizations based on linear tensor‐product and simplex‐type finite elements. The resulting discretizations are continuous in space, and continuous or discontinuous in time. In a first test run, all four methods are applied to a linear scalar advection‐diffusion model problem. Then, the convergence properties of the time‐discontinuous space‐time finite element discretizations are studied in numerical experiments. Advection velocity and diffusion coefficient are varied, such that the parabolic case of pure diffusion (heat equation), as well as, the hyperbolic case of pure advection (transport equation) are included in the study. For each model parameter set, the L2$$ {L}_2 $$ error at the final time is computed for spatial and temporal element lengths ranging over several orders of magnitude to allow for an individual evaluation of the methods' spatial, temporal, and space‐time accuracy. In the parabolic case, particular attention is paid to the influence of time‐dependent boundary conditions. Key findings include a spatial accuracy of second order and a temporal accuracy between second and third order. The temporal accuracy tends toward third order depending on how advection‐dominated the test case is, on the choice of the specific discretization method, and on the time‐(in)dependence and treatment of the boundary conditions. Additionally, the potential of time‐continuous simplex space‐time finite elements for heat flux computations is demonstrated with a piston ring pack test case and a subtractive manufacturing test case.

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