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H2020

ExaHyPE Report Summary

Project ID: 671698
Funded under: H2020-EU.1.2.2.

Periodic Reporting for period 1 - ExaHyPE (An Exascale Hyperbolic PDE Engine)

Reporting period: 2015-10-01 to 2016-09-30

Summary of the context and overall objectives of the project

The project ExaHyPE develops an exascale-ready engine to solve hyperbolic systems of partial differential equations (PDE) on next-generation supercomputers. Hyperbolic PDE systems, often resulting from conservation laws, are the basis to model a wide range of phenomena and processes in science and engineering. ExaHyPE particularly addresses problems in Seismology (seismic wave propagation, as for example for earthquake simulation) and Astrophysics (relativistic equations to model systems of neutron stars collapsing to a black hole and generating gravitational waves). However, as an “engine” the developed simulation software package is intended to enable groups of computational scientists to realize grand challenge simulations in short time for other fields, such as areas of fluid dynamics, various wave propagation problems, and many others.

Simulation software for exascale supercomputers will face novel challenges, as it will have to deal with questions of energy efficiency and fluctuating performance (due to measures to improve resiliency), and will need to scale on machines with multiple levels of parallelism including tens-of-millions of compute cores and possible hundreds-of-millions of parallel threads. ExaHyPE addresses these challenges by further developing a novel, high-order discontinuous-Galerkin Finite-Element-type discretisation scheme combined with a high-order communication-avoiding time stepping scheme (ADER-DG). At shocks and other regions where high-order accuracy is prevented by the characteristics of the solution, the solver falls back to a robust Finite-Volume limiting procedure. The resulting scheme is expected to ideally fit future supercomputing architectures that require high arithmetic intensity and as-few-as-possible communication. To parallelise on millions of possibly heterogeneous compute core, we use tree-structured approaches for discretisation, for which we will develop decentralised agile load balancing approaches.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

In the first project year, a prototype of the ExaHyPE engine was developed that brings together the implementation of the high-order numerical scheme with the tree-structured Cartesian meshing paradigm. The prototype offers a flexible programming interface to realise arbitrary hyperbolic PDE systems, and offers hybrid shared&distributed parallelism transparent to the user. The protoype was evaluated on several benchmark scenarios from seismology, astrophysics, and general fluid dynamics. A code generator was developed that provides optimised compute kernels tailored to the specific PDE system and discretisation order. Further work packages prepare the envisaged grand challenge simulations. For example, a truly hyperbolic first-order PDE system of the Einstein equations, featuring magneto-hydrodynamic fluid dynamics in relativistic space-time, was developed and solved with the high-order ADER-DG method.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

In the first project year, a key objective in ExaHyPE was to design the PDE engine for best flexibility and usability, to allow widespread use of the engine in the Computational Science and Engineering community. In addition, the focus was on integrating algorithmic and software components. While straightforward implementations of the adopted ADER-DG with Finite-Volume limiting scheme may require up to six traversals of the discretisation grid, the developed engine requires only a single amortised traversal for the entire scheme, which also minimises the required data transfers between memory and CPU. In addition, the ExaHyPE engine can compress degrees of freedom on-the-fly to lower accuracy (single vs. double precision, e.g.), further decreasing the memory requirements. First tests with the developed code generator have shown performance up to 35% of peak on modern CPU architectures. Both the engine and the code generator have been tested on Intel’s novel Xeon Phi processor (Knights Landing). After integration of the code generator and based on the single-traversal implementation, substantial reduction in time-to-solution of the ADER-DG scheme is expected.

The developed numerical models based on the Z4 and CCZ4 formulations of the Einstein equations are completely novel in the sense that they are written as first-order partial differential equations and strictly hyperbolic. While these properties are necessary to make them solvable via the ExaHyPE engine, first tests also show superior accuracy of the models.

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