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Journal articles on the topic 'ADER-DG'

Author: Grafiati

Published: 10 March 2023

Last updated: 31 July 2025

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1

Dumbser, Michael, Francesco Fambri, Maurizio Tavelli, Michael Bader, and Tobias Weinzierl. "Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine." Axioms 7, no. 3 (2018): 63. http://dx.doi.org/10.3390/axioms7030063.

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In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication-avoiding and cache-blocking, and are furthermore very well-suited for vectorization, and so they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the

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2

Wenk, S., C. Pelties, H. Igel, and M. Käser. "Regional wave propagation using the discontinuous Galerkin method." Solid Earth Discussions 4, no. 2 (2012): 1129–64. http://dx.doi.org/10.5194/sed-4-1129-2012.

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Abstract. We present an application of the discontinuous Galerkin (DG) method to regional wave propagation. The method makes use of unstructured tetrahedral meshes, combined with a time integration scheme solving the arbitrary high-order derivative (ADER) Riemann problem. The ADER-DG method is high-order accurate in space and time, beneficial for reliable simulations of high-frequency wavefields over long propagation distances. Due to the ease with which tetrahedral grids can be adapted to complex geometries, undulating topography of the Earth's surface and interior interfaces can be readily i

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3

Wenk, S., C. Pelties, H. Igel, and M. Käser. "Regional wave propagation using the discontinuous Galerkin method." Solid Earth 4, no. 1 (2013): 43–57. http://dx.doi.org/10.5194/se-4-43-2013.

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Abstract. We present an application of the discontinuous Galerkin (DG) method to regional wave propagation. The method makes use of unstructured tetrahedral meshes, combined with a time integration scheme solving the arbitrary high-order derivative (ADER) Riemann problem. This ADER-DG method is high-order accurate in space and time, beneficial for reliable simulations of high-frequency wavefields over long propagation distances. Due to the ease with which tetrahedral grids can be adapted to complex geometries, undulating topography of the Earth's surface and interior interfaces can be readily

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4

Pelties, C., A. A. Gabriel, and J. P. Ampuero. "Verification of an ADER-DG method for complex dynamic rupture problems." Geoscientific Model Development 7, no. 3 (2014): 847–66. http://dx.doi.org/10.5194/gmd-7-847-2014.

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Abstract. We present results of thorough benchmarking of an arbitrary high-order derivative discontinuous Galerkin (ADER-DG) method on unstructured meshes for advanced earthquake dynamic rupture problems. We verify the method by comparison to well-established numerical methods in a series of verification exercises, including dipping and branching fault geometries, heterogeneous initial conditions, bimaterial interfaces and several rate-and-state friction laws. We show that the combination of meshing flexibility and high-order accuracy of the ADER-DG method makes it a competitive tool to study

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5

Pelties, C., A. A. Gabriel, and J. P. Ampuero. "Verification of an ADER-DG method for complex dynamic rupture problems." Geoscientific Model Development Discussions 6, no. 4 (2013): 5981–6034. http://dx.doi.org/10.5194/gmdd-6-5981-2013.

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Abstract. We present thorough benchmarking of an arbitrary high-order derivative Discontinuous Galerkin (ADER-DG) method on unstructured meshes for advanced earthquake dynamic rupture problems. We validate the method in comparison to well-established numerical methods in a series of verification exercises, including dipping and branching fault geometries, heterogeneous initial conditions, bi-material cases and several rate-and-state friction constitutive laws. We show that the combination of meshing flexibility and high-order accuracy of the ADER-DG method makes it a competitive tool to study

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6

Gaburro, Elena, Philipp Öffner, Mario Ricchiuto, and Davide Torlo. "High order entropy preserving ADER-DG schemes." Applied Mathematics and Computation 440 (March 2023): 127644. http://dx.doi.org/10.1016/j.amc.2022.127644.

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7

Zhao, Xiaoxu, Baining Wang, Gang Li, and Shouguo Qian. "A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations." Mathematics 12, no. 16 (2024): 2601. http://dx.doi.org/10.3390/math12162601.

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In this article, we propose a new path-conservative discontinuous Galerkin (DG) method to solve non-conservative hyperbolic partial differential equations (PDEs). In particular, the method here applies the one-stage ADER (Arbitrary DERivatives in space and time) approach to fulfill the temporal discretization. In addition, this method uses the differential transformation (DT) procedure rather than the traditional Cauchy–Kowalewski (CK) procedure to achieve the local temporal evolution. Compared with the classical ADER methods, the current method is free of solving generalized Riemann problems

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8

Guerrero Fernández, Ernesto, Cipriano Escalante, and Manuel J. Castro Díaz. "Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws." Mathematics 10, no. 1 (2021): 15. http://dx.doi.org/10.3390/math10010015.

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This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge–Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurio

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9

Popov, I. S. "Space–time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting for multidimensional detonation waves simulation." Computers & Fluids 284 (November 2024): 106425. http://dx.doi.org/10.1016/j.compfluid.2024.106425.

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10

Castro, C. E., J. Behrens, and C. Pelties. "CUDA-C implementation of the ADER-DG method for linear hyperbolic PDEs." Geoscientific Model Development Discussions 6, no. 3 (2013): 3743–86. http://dx.doi.org/10.5194/gmdd-6-3743-2013.

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Abstract. We implement the ADER-DG numerical method using the CUDA-C language to run the code in a Graphic Processing Unit (GPU). We focus on solving linear hyperbolic partial differential equations where the method can be expressed as a combination of precomputed matrix multiplications becoming a good candidate to be used on the GPU hardware. Moreover, the method is arbitrarily high-order involving intensive work on local data, a property that is also beneficial for the target hardware. We compare our GPU implementation against CPU versions of the same method observing similar convergence pro

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11

Charrier, Dominic E., Benjamin Hazelwood, Ekaterina Tutlyaeva, et al. "Studies on the energy and deep memory behaviour of a cache-oblivious, task-based hyperbolic PDE solver." International Journal of High Performance Computing Applications 33, no. 5 (2019): 973–86. http://dx.doi.org/10.1177/1094342019842645.

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We study the performance behaviour of a seismic simulation using the ExaHyPE engine with a specific focus on memory characteristics and energy needs. ExaHyPE combines dynamically adaptive mesh refinement (AMR) with ADER-DG. It is parallelized using tasks, and it is cache efficient. AMR plus ADER-DG yields a task graph which is highly dynamic in nature and comprises both arithmetically expensive tasks and tasks which challenge the memory’s latency. The expensive tasks and thus the whole code benefit from AVX vectorization, although we suffer from memory access bursts. A frequency reduction of t

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12

van Gestel, Robert A. M., Martijn J. H. Anthonissen, Jan H. M. ten Thije Boonkkamp, and Wilbert L. IJzerman. "A hybrid semi-Lagrangian DG and ADER-DG solver on a moving mesh for Liouville's equation of geometrical optics." Journal of Computational Physics 498 (February 2024): 112655. http://dx.doi.org/10.1016/j.jcp.2023.112655.

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13

Castro, Cristóbal E., Jörn Behrens, and Christian Pelties. "Optimization of the ADER-DG method in GPU applied to linear hyperbolic PDEs." International Journal for Numerical Methods in Fluids 81, no. 4 (2015): 195–219. http://dx.doi.org/10.1002/fld.4179.

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14

Fambri, Francesco, Michael Dumbser, and Olindo Zanotti. "Space–time adaptive ADER-DG schemes for dissipative flows: Compressible Navier–Stokes and resistive MHD equations." Computer Physics Communications 220 (November 2017): 297–318. http://dx.doi.org/10.1016/j.cpc.2017.08.001.

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15

Wolf, Sebastian, Martin Galis, Carsten Uphoff, et al. "An efficient ADER-DG local time stepping scheme for 3D HPC simulation of seismic waves in poroelastic media." Journal of Computational Physics 455 (April 2022): 110886. http://dx.doi.org/10.1016/j.jcp.2021.110886.

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16

Rannabauer, Leonhard, Michael Dumbser, and Michael Bader. "ADER-DG with a-posteriori finite-volume limiting to simulate tsunamis in a parallel adaptive mesh refinement framework." Computers & Fluids 173 (September 2018): 299–306. http://dx.doi.org/10.1016/j.compfluid.2018.01.031.

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17

Bassi, C., S. Busto, and M. Dumbser. "High order ADER-DG schemes for the simulation of linear seismic waves induced by nonlinear dispersive free-surface water waves." Applied Numerical Mathematics 158 (December 2020): 236–63. http://dx.doi.org/10.1016/j.apnum.2020.08.005.

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18

Popov, Ivan S. "Efficient implementation of space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows." Journal of Physics: Conference Series 1740 (January 2021): 012059. http://dx.doi.org/10.1088/1742-6596/1740/1/012059.

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19

Popov, Ivan S. "High Order ADER-DG Method with Local DG Predictor for Solutions of Differential-Algebraic Systems of Equations." Journal of Scientific Computing 102, no. 2 (2025). https://doi.org/10.1007/s10915-024-02769-x.

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20

Fernández, E. Guerrero, M. J. Castro Díaz, M. Dumbser, and T. Morales de Luna. "An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density." Journal of Scientific Computing 90, no. 1 (2021). http://dx.doi.org/10.1007/s10915-021-01734-2.

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AbstractIn this work, we present a novel numerical discretization of a variable pressure multilayer shallow water model. The model can be written as a hyperbolic PDE system and allows the simulation of density driven gravity currents in a shallow water framework. The proposed discretization consists in an unlimited arbitrary high order accurate (ADER) Discontinuous Galerkin (DG) method, which is then limited with the MOOD paradigm using an a posteriori subcell finite volume limiter. The resulting numerical scheme is arbitrary high order accurate in space and time for smooth solutions and does

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21

Dorozhinskii, Ravil, Gonzalo Brito Gadeschi, and Michael Bader. "Fused GEMMs towards an efficient GPU implementation of the ADER‐DG method in SeisSol." Concurrency and Computation: Practice and Experience, February 13, 2024. http://dx.doi.org/10.1002/cpe.8037.

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SummaryThis study shows how GPU performance of the ADER discontinuous Galerkin method in SeisSol (an earthquake simulation software) can be further improved while preserving its original design that ensures high CPU performance. We introduce a new code generator (“ChainForge”) that fuses subsequent batched matrix multiplications (“GEMMs”) into a single GPU kernel, holding intermediate results in shared memory as long as necessary. The generator operates as an external module linked against SeisSol's domain specific language YATeTo and, as a result, the original SeisSol source code remains main

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22

Popov, Ivan S. "Arbitrary High Order ADER-DG Method with Local DG Predictor for Solutions of Initial Value Problems for Systems of First-Order Ordinary Differential Equations." Journal of Scientific Computing 100, no. 1 (2024). http://dx.doi.org/10.1007/s10915-024-02578-2.

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23

Río-Martín, Laura, and Michael Dumbser. "High-Order ADER Discontinuous Galerkin Schemes for a Symmetric Hyperbolic Model of Compressible Barotropic Two-Fluid Flows." Communications on Applied Mathematics and Computation, November 30, 2023. http://dx.doi.org/10.1007/s42967-023-00313-6.

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AbstractThis paper presents a high-order discontinuous Galerkin (DG) finite-element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) model of compressible two-phase flow, introduced by Romenski et al. in [59, 62], in multiple space dimensions. In the absence of algebraic source terms, the model is endowed with a curl constraint on the relative velocity field. In this paper, the hyperbolicity of the system is studied for the first time in the multidimensional case, showing that the original model is only weakly hyperbolic in

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24

Popov, I. S. "Space-Time Adaptive ADER-DG Finite Element Method with LST-DG Predictor and a posteriori Sub-cell WENO Finite-Volume Limiting for Simulation of Non-stationary Compressible Multicomponent Reactive Flows." Journal of Scientific Computing 95, no. 2 (2023). http://dx.doi.org/10.1007/s10915-023-02164-y.

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25

Busto, Saray, Michael Dumbser, Sergey Gavrilyuk, and Kseniya Ivanova. "On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows." Journal of Scientific Computing 88, no. 1 (2021). http://dx.doi.org/10.1007/s10915-021-01521-z.

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AbstractIn this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this

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26

Gaburro, Elena, and Michael Dumbser. "A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics." Journal of Scientific Computing 86, no. 3 (2021). http://dx.doi.org/10.1007/s10915-020-01405-8.

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AbstractIn this work, we consider the general family of the so called ADER$$P_NP_M$$PNPMschemes for the numerical solution of hyperbolic partial differential equations witharbitraryhigh order of accuracy in space and time. The family of one-step$$P_NP_M$$PNPMschemes was introduced in Dumbser (J Comput Phys 227:8209–8253, 2008) and represents a unified framework for classical high order Finite Volume (FV) schemes ($$N=0$$N=0), the usual Discontinuous Galerkin (DG) methods ($$N=M$$N=M), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degreeMis applied

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