| 1. |
- Abgrall, Remi, et al.
(author)
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Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability
- 2023
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In: Communications on Applied Mathematics and Computation. - : Springer. - 2096-6385 .- 2661-8893. ; 5:2, s. 573-595
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Journal article (peer-reviewed)abstract
- In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. By switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.
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| 2. |
- Dedner, Andreas, et al.
(author)
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Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods
- 2022
-
In: Communications on Applied Mathematics and Computation. - : Springer Science and Business Media LLC. - 2096-6385 .- 2661-8893. ; 4:2, s. 657-696
-
Journal article (peer-reviewed)abstract
- This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efficient implementations of the discontinuous Galerkin (DG) method for solving a wide range of nonlinear partial differential equations (PDEs). Although the C++ interfaces of Dune-Fem-DG are highly flexible and customizable, a solid knowledge of C++ is necessary to make use of this powerful tool. With this work, easier user interfaces based on Python and the unified form language are provided to open Dune-Fem-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and first-order hyperbolic PDEs.
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| 3. |
- Ranocha, Hendrik, et al.
(author)
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On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics
- 2023
-
In: Communications on Applied Mathematics and Computation. - : SPRINGERNATURE. - 2096-6385 .- 2661-8893.
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Journal article (peer-reviewed)abstract
- We study a temporal step size control of explicit Runge-Kutta (RK) methods for compressible computational fluid dynamics (CFD), including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations. We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy (CFL) number. Our numerical examples show that the error-based step size control is easy to use, robust, and efficient, e.g., for (initial) transient periods, complex geometries, nonlinear shock capturing approaches, and schemes that use nonlinear entropy projections. We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases, the open source Julia packages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.
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| 4. |
- Ryan, Jennifer K.
(author)
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Capitalizing on Superconvergence for More Accurate Multi-Resolution Discontinuous Galerkin Methods
- 2022
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In: Communications on Applied Mathematics and Computation. - : Springer Nature. - 2096-6385 .- 2661-8893. ; 4:2, s. 417-436
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Journal article (peer-reviewed)abstract
- This article focuses on exploiting superconvergence to obtain more accurate multi-resolution analysis. Specifically, we concentrate on enhancing the quality of passing of information between scales by implementing the Smoothness-Increasing Accuracy-Conserving (SIAC) filtering combined with multi-wavelets. This allows for a more accurate approximation when passing information between meshes of different resolutions. Although this article presents the details of the SIAC filter using the standard discontinuous Galerkin method, these techniques are easily extendable to other types of data.
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