Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints

Voulis, Igor; Reusken, Arnold

doi:10.1515/jnma-2018-0013

Cited by 9 publications

(8 citation statements)

References 33 publications

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“…The presented expansion may be used for the construction of novel numerical methods. For this, the approach needs to be combined with integration schemes for PDAEs of parabolic type such as splitting schemes [AO17], Runge-Kutta methods [AZ18a,Zim21], discontinuous Galerkin methods [VR19], or exponential integrators [AZ20,Zim21].…”

Section: Discussionmentioning

confidence: 99%

Singular perturbation results for linear partial differential-algebraic equations of hyperbolic type

Altmann¹,

Zimmer²

2021

Preprint

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We consider constrained partial differential equations of hyperbolic type with a small parameter ε > 0, which turn parabolic in the limit case, i. e., for ε = 0. The well-posedness of the resulting systems is discussed and the corresponding solutions are compared in terms of the parameter ε. For the analysis, we consider the system equations as partial differential-algebraic equation based on the variational formulation of the problem. Depending on the particular choice of the initial data, we reach firstand second-order estimates. Optimality of the lower-order estimates for general initial data is shown numerically.

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Section: Discussionmentioning

confidence: 99%

Singular perturbation results for linear partial differential-algebraic equations of hyperbolic type

Altmann¹,

Zimmer²

2021

Preprint

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“…The result numerically confirms that temporal superconvergence (as stated in Equation 22) can also be obtained for time-dependent boundary conditions with a proper treatment. 32 For D-SST with b(t l ), only quadratic temporal convergence is observed. The lower convergence order of the D-SST method with treatment of the time-dependent boundary conditions hints at the fact that superconvergence of the D-PST method is linked to the tensor-product structure of the discretization.…”

Section: (A) (B)mentioning

confidence: 96%

“…31 The influence of linear constraints, for example, time-dependent Dirichlet boundary conditions, on discontinuous Galerkin time discretization methods for parabolic problems is treated by Voulis and Reusken. 32…”

Section: Literature Reviewmentioning

confidence: 99%

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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“…The aim of this paper is to derive time discretization schemes for the given class of PDAEs. Recently, time discretization schemes have been analyzed for the parabolic case (ε = 0) including Runge-Kutta methods [AZ18] and discontinuous Galerkin methods [VR18]. In general, one may say that the construction and analysis of numerical schemes need a combination of methods known from time-dependent PDEs, see, e.g., [LO95,ET10], as well as strategies coming form the theory of differential-algebraic equations (DAE), cf.…”

Section: Introductionmentioning

confidence: 99%

Time discretization of nonlinear hyperbolic systems on networks

Altmann

Zimmer

2019

Proc Appl Math and Mech

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Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints

Cited by 9 publications

References 33 publications

Singular perturbation results for linear partial differential-algebraic equations of hyperbolic type

Singular perturbation results for linear partial differential-algebraic equations of hyperbolic type

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

Time discretization of nonlinear hyperbolic systems on networks

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