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A stabilized DG cut cell method for discretizing the linear transport equation

Christian Engwer, Sandra May, Andreas Nüßing, Florian Streitbürger, Technische Universität Dortmund, Technische Universität Dortmund
We present new stabilization terms for solving the linear transport equation on a cut cell mesh using the discontinuous Galerkin (DG) method in two dimensions with piecewise linear polynomials.  ...  Using a method of lines approach, we start with a standard upwind DG discretization for the background mesh and add penalty terms that stabilize the solution on small cut cells in a conservative way.  ...  For the cut cell discretization we rely on the dune-udg package [20] and its integration with dune-pdelab [8] . The dune-udg module was originally developed for the unfitted DG method [6] .  ... 
doi:10.17877/de290r-20076 fatcat:l6iqc6o7ejczdifrxuftfxhdai

A Generalized Eulerian-Lagrangian Discontinuous Galerkin Method for Transport Problems [article]

Xue Hong, Jing-Mei Qiu
2021 arXiv   pre-print
The method is a generalization of the Eulerian-Lagrangian (EL) DG method for transport problems proposed in [arXiv preprint arXiv: 2002.02930 (2020)], which tracks solution along approximations to characteristics  ...  Numerical results on 1D and 2D linear transport problems are presented to demonstrate great properties of the GEL DG method.  ...  Thus the stability result of ALE DG method for linear conservation laws [28] can be directly applied to assess the stability property of fully discrete EL RK DG scheme.  ... 
arXiv:2102.11383v1 fatcat:uaqubuzq3vhkhfueu6mbvwgpei

Numerical solution of steady-state groundwater flow and solute transport problems: Discontinuous Galerkin based methods compared to the Streamline Diffusion approach [article]

A.Q.T. Ngo, P. Bastian, O. Ippisch
2014 arXiv   pre-print
Furthermore, we illustrate an efficient way of solving the linear system arising from the DG discretization.  ...  For an interior penalty discontinuous Galerkin (DG) discretization, we present a h-adaptive refinement strategy and, alternatively, a new efficient approach for reducing numerical under- and overshoots  ...  Acknowledgements This study has been funded by the Baden-Württemberg Stiftung in its high-performance computing program, contract HPC-8.  ... 
arXiv:1411.1432v1 fatcat:syuzyjdg2rbhfistfvp526ltza

Monotonicity considerations for stabilized DG cut cell schemes for the unsteady advection equation [article]

Florian Streitbürger and Christian Engwer and Sandra May and Andreas Nüßing
2019 arXiv   pre-print
In a recent preprint [arXiv:1906.05642], we propose penalty terms for stabilizing a DG space discretization to overcome this issue for the unsteady linear advection equation.  ...  For solving unsteady hyperbolic conservation laws on cut cell meshes, the so called small cell problem is a big issue: one would like to use a time step that is chosen with respect to the background mesh  ...  A stabilized DG cut cell scheme for the unsteady advection equation We consider the time dependent linear advection problem on a cut cell mesh.  ... 
arXiv:1912.11933v1 fatcat:smtglr667jeubgbctxc74rv6fm

DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension [article]

Sandra May, Florian Streitbürger
2021 arXiv   pre-print
When applied on a cut cell mesh with a time step length that is appropriate for the size of the larger background cells, one encounters stability issues.  ...  The DoD stabilization consists of penalty terms that are designed to address these problems by redistributing mass between the inflow and outflow neighbors of small cut cells in a physical way.  ...  Acknowledgments The authors would like to thank Christian Engwer, Andrew Giuliani, and Tim Mitchell for helpful discussions.  ... 
arXiv:2107.03689v1 fatcat:mkqonipaandgvb6geo6yoh6tdu

Fast computation of multiphase flow in porous media by implicit discontinuous Galerkin schemes with optimal ordering of elements

Jostein R. Natvig, Knut-Andreas Lie
2008 Journal of Computational Physics  
To advance the solution one time step, one must solve a discrete system of nonlinear equations.  ...  In particular, the first-order version of the method may be at least as efficient as modern streamline methods when accuracy requirements or the dynamics of the flow allow for large implicit time steps  ...  Acknowledgement We would like to thank Vegard Kippe for the streamline simulations in Case 3, and Ola I. Røe for the simulation of fractured media using triangular elements.  ... 
doi:10.1016/ fatcat:uk7uk57jenf7hbqh5plzfg7s3a

Discontinuous Galerkin method for incompressible two-phase flows [article]

Janick Thomas Gerstenberger, Samuel Burbulla, Dietmar Kröner
2020 arXiv   pre-print
The scheme is especially suitable for two-phase flow when used with a piecewise-linear interface construction (PLIC) volume-of-fluid (VoF) method and cut-cell quadratures.  ...  The scheme does not need penalty parameters and satisfies the discrete continuity equation exactly.  ...  Acknowledgements The first author would like to thank Tobias Malkmus for providing the initial code base for the fluid solver and explaining its subtleties, Martin Nolte for his ideas and help in modifying  ... 
arXiv:2003.12373v1 fatcat:5rghegdhozfkxa6uqrv4mf3gmm

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations

Yingda Cheng, Irene M. Gamba, Jennifer Proft
2012 Mathematics of Computation  
A discussion of the standard semi-discrete DG schemes for the BTE are included as a foundation for the stability and error estimates for this new scheme.  ...  We develop a high-order positivity-preserving discontinuous Galerkin (DG) scheme for linear Vlasov-Boltzmann transport equations (Vlasov-BTE) under the action of quadratically confined electrostatic potentials  ...  The authors thank Chi-Wang Shu and Xiangxiong Zhang for discussions about the maximum-principle-satisfying schemes for conservation laws.  ... 
doi:10.1090/s0025-5718-2011-02504-4 fatcat:bbhw5ndsyjb4ve7mvbdw4qt72q

Implicit finite volume and discontinuous Galerkin methods for multicomponent flow in unstructured 3D fractured porous media

Joachim Moortgat, Mohammad Amin Amooie, Mohamad Reza Soltanian
2016 Advances in Water Resources  
to the pore volume of discrete fracture grid cells.  ...  A lowest-order implicit finite volume (FV) transport update is also developed for the same grid types. The implicit methods are compared to an Implicit-Pressure-Explicit-Composition (IMPEC) scheme.  ...  weak form of the transport equation in the DG discretization uses a partial integration that results in one volume integral that involves only the variables inside the grid-cell, F (c m i , q E ), and  ... 
doi:10.1016/j.advwatres.2016.08.007 fatcat:udee3bylajh6hk46geefyohkfe

A New hp-Adaptive DG Scheme for Conservation Laws Based on Error Control [chapter]

A. Dedner, M. Ohlberger
2008 Hyperbolic Problems: Theory, Numerics, Applications  
The hp-adaptivity of the scheme is based on a rigorous a posteriori error estimate for a generalized class of Discontinuous Galerkin schemes presented in [DMO06] .  ...  Numerical experiments demonstrate the efficiency and stability of the scheme in multiple space dimensions.  ...  Linear advection: As a first numerical example we look at the linear transport equation ∂ t u + a∇u = 0 , u(·, 0) = u 0 (·) with the constant transport velocity a = (2.25, 0.22).  ... 
doi:10.1007/978-3-540-75712-2_15 fatcat:knyq3d6rgfewriujt773x3fmry

A Positive and Stable L2-minimization Based Moment Method for the Boltzmann Equation of Gas dynamics [article]

Neeraj Sarna
2020 arXiv   pre-print
We consider the method-of-moments approach to solve the Boltzmann equation of rarefied gas dynamics, which results in the following moment-closure problem.  ...  We show that a (Courant-Friedrichs-Lewy) CFL-type condition ensures both the feasibility of the optimization problem and the L2-stability of the moment approximation.  ...  This makes the above evolution equation a space-time discretization of a system of decoupled linear advection equations given as ∂ t W (f ) + Ξ∂ x W (f ) = 0.  ... 
arXiv:2009.11376v1 fatcat:lwbjeh632jctjl5w34txygucne

High-resolution finite element methods for 3D simulation of compositionally triggered instabilities in porous media

Ebrahim Shahraeeni, Joachim Moortgat, Abbas Firoozabadi
2015 Computational Geosciences  
This is based on a combination of the mixed hybrid finite element (MHFE) method for total fluid velocity and discontinuous Galerkin (DG) method for the species transport.  ...  To achieve such a high resolution, we present higherorder 3D finite element methods for the simulation of fully compositional, three-phase and multi-component flow.  ...  Acknowledgments This work is supported by the members companies of the Reservoir Engineering Research Institute (RERI). Their support is greatly appreciated.  ... 
doi:10.1007/s10596-015-9501-z fatcat:f4vmjoqf3nc2jifft5ehcbyml4

High order modal Discontinuous Galerkin Implicit-Explicit Runge Kutta and Linear Multistep schemes for the Boltzmann model on general polygonal meshes [article]

Walter Boscheri, Giacomo Dimarco
2021 arXiv   pre-print
The solution of the Boltzmann collision operator is obtained through fast spectral methods, while the transport term in the governing equations is discretized relying on an explicit shock-capturing DG  ...  To that aim, we develop modal Discontinuous Galerkin (DG) Implicit-Explicit Runge Kutta schemes (DG-IMEX-RK) and Implicit-Explicit Linear Multistep Methods based on Backward-Finite-Differences (DG-IMEX-BDF  ...  "Innovative numerical methods for evolutionary partial differential equations and applications").  ... 
arXiv:2107.11101v1 fatcat:fozqcazt7jgd5odbdhj6uekxje

High-order Finite Difference and Finite Volume WENO Schemes and Discontinuous Galerkin Methods for CFD

Chi-Wang Shu
2003 International journal of computational fluid dynamics (Print)  
discontinuous Galerkin DG nite element methods.  ...  We summarize the main features of these methods, from a practical user's point of view, indicate their applicability and relative strength, and show a few selected numerical examples to demonstrate their  ...  The rst discontinuous Galerkin method was introduced in 1973 by Reed and Hill 35 , in the framework of neutron transport, i.e. a time independent linear hyperbolic equation.  ... 
doi:10.1080/1061856031000104851 fatcat:ysiswr2x4nal3nuinmuq5wvui4

A semi-Lagrangian discontinuous Galerkin (DG) – local DG method for solving convection-diffusion-reaction equations [article]

Mingchang Ding, Xiaofeng Cai, Wei Guo, Jing-Mei Qiu
2019 arXiv   pre-print
The method generalizes our previous work on developing the SLDG method for transport equations (J. Sci.  ...  In this paper, we propose an efficient high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for solving linear convection-diffusion-reaction equations.  ...  The DG discretization approach is a class of finite element methods that use piecewise continuous approximations and enjoy many attractive computational advantages for transport dominant problems.  ... 
arXiv:1907.06117v1 fatcat:mua6blxxbraejlh4aoshzuzlqq
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