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The discontinuous Galerkin (DG) algorithm is a representative high order method in Computational Fluid Dynamics (CFD) area which possesses considerable mathematical advantages such as high resolution, low dissipation, and dispersion.…
We present a sequence of optimizations to the performance-critical compute kernels of the high-order discontinuous Galerkin solver of the hyperbolic PDE engine ExaHyPE -- successively tackling bottlenecks due to SIMD operations, cache…
A compiler approach for generating low-level computer code from high-level input for discontinuous Galerkin finite element forms is presented. The input language mirrors conventional mathematical notation, and the compiler generates…
Leveraging the SIMD capability of modern CPU architectures is mandatory to take full benefit of their increasing performance. To exploit this feature, binary executables must be explicitly vectorized by the developers or an automatic…
Achieving a substantial part of peak performance on todays and future high-performance computing systems is a major challenge for simulation codes. In this paper we address this question in the context of the numerical solution of partial…
Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of…
The third paper in our series on open source MATLAB / GNU Octave implementation of the discontinuous Galerkin (DG) method(s) focuses on a hybridized formulation. The main aim of this ongoing work is to develop rapid prototyping techniques…
In this paper we discuss a new and very efficient implementation of high order accurate ADER discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad…
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 non linear partial differential…
We describe a high order technique to generate quadrilateral decompositions and meshes for complex two dimensional domains using spectral elements in a field guided procedure. Inspired by cross field methods, we never actually compute…
The shallow water equations (SWE) are a commonly used model to study tsunamis, tides, and coastal ocean circulation. However, there exist various approaches to discretize and solve them efficiently. Which of them is best for a certain…
We present a novel high-order accurate nodal discontinuous Galerkin (DG) method for solving nonlinear hyperbolic systems of partial differential equations (PDEs) on fully unstructured three-dimensional polyhedral meshes. A mesh generator is…
The implementation of discontinuous Galerkin finite element methods (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist…
Vectorization via Single Instruction, Multiple Data (SIMD) architectures is a cornerstone of high-performance computing. To fully exploit hardware potential, developers often resort to explicit vectorization using intrinsics, as…
We introduce a new family of discontinuous Galerkin (DG) finite element schemes for the discretization of first order systems of hyperbolic partial differential equations (PDE) on unstructured simplex meshes in two and three space…
The numerical solution of partial differential equations is at the heart of many grand challenges in supercomputing. Solvers based on high-order discontinuous Galerkin (DG) discretisation have been shown to scale on large supercomputers…
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and…
We present a unified algorithmic framework for the numerical solution, constrained optimization, and physics-informed learning of PDEs with a variational structure. Our framework is based on a Galerkin discretization of the underlying…
This is the second in a series of papers on implementing a discontinuous Galerkin (DG) method as an open source Matlab / GNU Octave toolbox. The intention of this ongoing project is to offer a rapid prototyping package for application…
We present a technique for automatically transforming kernel-based computations in disparate, nested loops into a fused, vectorized form that can reduce intermediate storage needs and lead to improved performance on contemporary hardware.…