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Journal articles on the topic 'ADER-DG'

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Author: Grafiati

Published: 10 March 2023

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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 (September 1, 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 numerical algorithm and show some applications to a set of hyperbolic equations with increasing levels of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes.

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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 (August 23, 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 implemented in the computational domain. The ADER-DG method is benchmarked for the accurate radiation of elastic waves excited by an explosive and a shear dislocation source. We compare real data measurements with synthetics of the 2009 L'Aquila event (central Italy). We take advantage of the geometrical flexibility of the approach to generate a European model composed of the 3-D EPcrust model, combined with the depth-dependent ak135 velocity model in the upper-mantle. The results confirm the applicability of the ADER-DG method for regional scale earthquake simulations, which provides an alternative to existing methodologies.

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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 (January 30, 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 implemented in the computational domain. The ADER-DG method is benchmarked for the accurate radiation of elastic waves excited by an explosive and a shear dislocation source. We compare real data measurements with synthetics of the 2009 L'Aquila event (central Italy). We take advantage of the geometrical flexibility of the approach to generate a European model composed of the 3-D EPcrust model, combined with the depth-dependent ak135 velocity model in the upper mantle. The results confirm the applicability of the ADER-DG method for regional scale earthquake simulations, which provides an alternative to existing methodologies.

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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 (May 13, 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 earthquake dynamics in geometrically complicated setups.

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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 (November 28, 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 earthquake dynamics in complicated setups.

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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

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 (December 21, 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 spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy.

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8

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 (July 13, 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 properties up to a threshold where the error remains fixed. This behaviour is in agreement with the CPU version but the threshold is larger that in the CPU case. We also observe a big difference when considering single and double precision where in the first case the threshold error is significantly larger. Finally, we did observe a speed up factor in computational time but this is relative to the specific test or benchmark problem.

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9

Charrier, Dominic E., Benjamin Hazelwood, Ekaterina Tutlyaeva, Michael Bader, Michael Dumbser, Andrey Kudryavtsev, Alexander Moskovsky, and Tobias Weinzierl. "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 (April 15, 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 the chip improves the code’s energy-to-solution. Yet, it does not mitigate burst effects. The bursts’ latency penalty becomes worse once we add Intel Optane technology, increase the core count significantly or make individual, computationally heavy tasks fall out of close caches. Thread overbooking to hide away these latency penalties becomes contra-productive with noninclusive caches as it destroys the cache and vectorization character. In cases where memory-intense and computationally expensive tasks overlap, ExaHyPE’s cache-oblivious implementation nevertheless can exploit deep, noninclusive, heterogeneous memory effectively, as main memory misses arise infrequently and slow down only few cores. We thus propose that upcoming supercomputing simulation codes with dynamic, inhomogeneous task graphs are actively supported by thread runtimes in intermixing tasks of different compute character, and we propose that future hardware actively allows codes to downclock the cores running particular task types.

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10

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 (October 25, 2015): 195–219. http://dx.doi.org/10.1002/fld.4179.

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11

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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12

Wolf, Sebastian, Martin Galis, Carsten Uphoff, Alice-Agnes Gabriel, Peter Moczo, David Gregor, and Michael Bader. "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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13

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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14

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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15

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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16

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 (December 17, 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 not destroy the natural subcell resolution inherent in the DG methods in the presence of strong gradients or discontinuities. A numerical strategy to preserve non-trivial stationary solutions is also discussed. The final method is very accurate in smooth regions even using coarse or very coarse meshes, as shown in the numerical simulations presented here. Finally, a comparison with a laboratory test, where empirical data are available, is also performed.

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17

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 (June 12, 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 paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

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18

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 (January 19, 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 over piecewise polynomial data of degreeNwith$$M>N$$M>N. In all cases with$$M \ge N > 0 $$M≥N>0the$$P_NP_M$$PNPMschemes arelinearin the sense of Godunov (Math. USSR Sbornik 47:271–306, 1959), thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of$$P_NP_M$$PNPMschemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order$$P_NP_M$$PNPMschemes, due to the use of a rather fine subgrid of$$2N+1$$2N+1subcells per space dimension. The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.

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