Variational Methods for Stable Time Discretization of First-Order Differential Equations

Becher, Simon; Matthies, Gunar; Wenzel, Dennis

doi:10.1007/978-3-319-97277-0_6

Cited by 10 publications

(20 citation statements)

References 1 publication

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“…Moreover, the orders of convergence do not decrease even if more than two postprocessing steps based on residuals are applied. The same behavior is observed for the methods Q 10 k -VTD 10 k , k = 0, . .…”

Section: Example 61supporting

confidence: 75%

“…While the first group shares its stability properties with the cGP method, the second group behaves like the dG method. These observations, presented in [10], imply that the considered methods are at least A-stable. This suggests that the whole family of methods is appropriate to handle stiff problems such as those arising, for example, from semidiscretizations of parabolic partial differential equations in space.…”

Section: Introductionmentioning

confidence: 64%

“…This suggests to choose large values for k, also cf. [10]. However, the larger k the more derivatives of the function F have to be provided.…”

Section: Remark 411mentioning

confidence: 99%

“…Our scope is to figure out how the approximation properties of these (external and internal) operators affect the error behavior of the numerical solution. Thereby our studies are not restricted to the dG or cGP method but cover the whole family of variational discretization methods recently introduced in [3,2]. The used assumptions are quite general and abstract in order enable the study of a wide variety of methods.…”

Section: Introductionmentioning

confidence: 99%

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Variational time discretizations of higher order and higher regularity

Becher

Matthies

2021

Bit Numer Math

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We consider a family of variational time discretizations that are generalizations of discontinuous Galerkin (dG) and continuous Galerkin–Petrov (cGP) methods. The family is characterized by two parameters. One describes the polynomial ansatz order while the other one is associated with the global smoothness that is ensured by higher order collocation conditions at both ends of the subintervals. Applied to Dahlquist’s stability problem, the presented methods provide the same stability properties as dG or cGP methods. Provided that suitable quadrature rules of Hermite type are used to evaluate the integrals in the variational conditions, the variational time discretization methods are connected to special collocation methods. For this case, we present error estimates, numerical experiments, and a computationally cheap postprocessing that allows to increase both the accuracy and the global smoothness by one order.

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Section: Example 61supporting

confidence: 75%

Section: Introductionmentioning

confidence: 64%

“…This suggests to choose large values for k, also cf. [10]. However, the larger k the more derivatives of the function F have to be provided.…”

Section: Remark 411mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Variational time discretizations of higher order and higher regularity

Becher

Matthies

2021

Bit Numer Math

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“…Continuously differentiable in time discrete solutions are obtained by the construction itself. The basic idea of the approach was proposed firstly in [4]. Optimal order error estimates for these schemes are proved in [1].…”

Section: Continuously Differentiable In Time Galerkin Approximationmentioning

confidence: 99%

Comparative study of continuously differentiable Galerkin time discretizations for the wave equation

Bause

Anselmann

2019

Proc Appl Math and Mech

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In this work we compare numerically two families of space-time finite element approximation schemes for wave equations. Continuously differentiable in time discrete solutions are provided by the schemes. The first family of schemes is based on a computationally cheap post-processing of the continuous in space and time finite element approximation of the wave equation. The second family combines the variational approximation in time by finite element techniques with the concepts of collocation methods. Both families show equal order of convergence. The corresponding error constants are reviewed here.

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Unified analysis for variational time discretizations of higher order and higher regularity applied to non-stiff ODEs

Becher

Matthies

2021

Numer Algor

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We present a unified analysis for a family of variational time discretization methods, including discontinuous Galerkin methods and continuous Galerkin–Petrov methods, applied to non-stiff initial value problems. Besides the well-definedness of the methods, the global error and superconvergence properties are analyzed under rather weak abstract assumptions which also allow considerations of a wide variety of quadrature formulas. Numerical experiments illustrate and support the theoretical results.

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Variational Methods for Stable Time Discretization of First-Order Differential Equations

Cited by 10 publications

References 1 publication

Variational time discretizations of higher order and higher regularity

Variational time discretizations of higher order and higher regularity

Comparative study of continuously differentiable Galerkin time discretizations for the wave equation

Unified analysis for variational time discretizations of higher order and higher regularity applied to non-stiff ODEs

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