Academic literature on the topic 'ADER-DG'

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.

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.

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.

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.

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.

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.

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.

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.

Looking into the structure, composition and behaviour of the Earth is one of the main goals of the seismic studies. Many geophysical problems — such as surface wave, group velocity and full waveform tomography , determination of mantle flows, gravity studies, source inversion — need plausible models as starting point for such studies. Crustal structure varies greatly over small scale length and has a strong effects on seismic waves. A priori models of the crust are thus often used to model seismic wave propagation at large distance and to account for shallow structure when imaging upper mantle structure. Focusing on forward earthquakes simulations, plausible crustal and mantle models are the first step to obtain realistic seismograms and results. Recent development in computer facilities and numerical methods — Spectral Element Method, ADER-DG method, Finite Difference method — enable to solve the wave equation in 3D complex media with high accuracy. These methods require a discrete representation of the investigation domain (mesh) through which we propagate wave. To model seismic wave propagation at the scale of a continent — i.e. signals travelling to stations a few hundred or thousand kilometers from the earthquake source — we have a problem connected to the detail and reliability of current models, that are sufficiently accurate when we look at the global scale, but often miss significant features at the scale of sedimentary basins and mountain ranges, that become very important as we zoom closer. Reliable and detailed information on these structures exist, for instance deriving from active-source studies, but are often not integrated in wide-area compilations such as desirable. At the European scale, it becomes clear that current crustal models are not adequate for modeling regional datasets with enough detail. The global model CRUST2.0 is frequently used for crustal correction and wave propagation, but its resolution is too low for continental-scale studies. Many other detailed information are available, but at different scales, with different information contents, and following different formats: this information needs to be merged into a larger-scale, coherent representation. The other important issue is that connected to the faithful implementation of a known structure in computational meshes used in forward simulations of wave propagation. The shallow crustal discontinuities indeed are difficult to represent, because of the small size of the shallower elements of the mesh that lead to a very short time step. In this study, I am mostly interested in addressing these two fundamental issues, i.e. how to retrieve a ’good’ crustal model for Europe, on the basis of existing knowledge, and how to best represent it for efficient, but accurate, numerical simulation of seismic wave propagation. In the first part (Chapter 1), we analyse the surface wave sensitivity to the crustal structure presenting an exercise, based on surface wave dispersion matching, to reparameterize CRUST2.0 global model in a simpler grid that can be considered equivalent to CRUST2.0 in modeling surface waves. The models is tested from a wave propagating point of view with SPECFEM-3D code. We collect all the informations available on the this region and we create a new comprehensive reference crustal model for the European plate (Chapter 2) that describes the complex structure of the Europe with higher resolution and more plausibility than previous models. However, we can improve the resolution of such large scale compilation: we collect tens of seismic lines in the East Alps region (Chapter 3) building up, applying a geostatistics technique, a complete regional crustal model of that area that was included in EPcrust. This would be an example in which new local models could be developed and integrated in the continental one. The results are available on www.bo.ingv.it/eurorem/EPcrust. Since new models are available, before starting a 3D implementation of the models in numerical methods, in Chapter 4 we quantitatively analyse in 2D the influence of the representation and uncertainties in the knowledge of crustal parameters on simulated wave field. We evaluate different synthetic test cases respect to the reference, analysing the frequency and source-receiver-distance dependence of our approximations. For the simulations, we use an high order ADER-DG scheme implemented in the SeisSol2D code able to honour the discontinuities in the crust with high fidelity. From a seismological point of view the next step after developing a model would be a validation of the model itself. In chapter 5, we go through a validation process of EPcrust. The main goal is to understand if our new model is able to give a better fit of the real data. We use the Spectral Element Method as implemented in SPECFEM3D-Globe. This choice would be a compromise between accuracy of the representation of the crustal structure and computational cost. The ADER-DG methods, well suited for an accurate representation of the sharp interface within the crust, is at the moment computationally too expensive for 3D simulations at continental scale. At the and of this thesis, we give a brief overview on methods and theory applied to obtain our results.
University of Bologna
Published
3.3. Geodinamica e struttura dell'interno della Terra
open