22 research outputs found
A hybridizable discontinuous Galerkin method for electromagnetics with a view on subsurface applications
Two Hybridizable Discontinuous Galerkin (HDG) schemes for the solution of
Maxwell's equations in the time domain are presented. The first method is based
on an electromagnetic diffusion equation, while the second is based on
Faraday's and Maxwell--Amp\`ere's laws. Both formulations include the diffusive
term depending on the conductivity of the medium. The three-dimensional
formulation of the electromagnetic diffusion equation in the framework of HDG
methods, the introduction of the conduction current term and the choice of the
electric field as hybrid variable in a mixed formulation are the key points of
the current study. Numerical results are provided for validation purposes and
convergence studies of spatial and temporal discretizations are carried out.
The test cases include both simulation in dielectric and conductive media
Efficient Explicit Time Stepping of High Order Discontinuous Galerkin Schemes for Waves
This work presents algorithms for the efficient implementation of
discontinuous Galerkin methods with explicit time stepping for acoustic wave
propagation on unstructured meshes of quadrilaterals or hexahedra. A crucial
step towards efficiency is to evaluate operators in a matrix-free way with
sum-factorization kernels. The method allows for general curved geometries and
variable coefficients. Temporal discretization is carried out by low-storage
explicit Runge-Kutta schemes and the arbitrary derivative (ADER) method. For
ADER, we propose a flexible basis change approach that combines cheap face
integrals with cell evaluation using collocated nodes and quadrature points.
Additionally, a degree reduction for the optimized cell evaluation is presented
to decrease the computational cost when evaluating higher order spatial
derivatives as required in ADER time stepping. We analyze and compare the
performance of state-of-the-art Runge-Kutta schemes and ADER time stepping with
the proposed optimizations. ADER involves fewer operations and additionally
reaches higher throughput by higher arithmetic intensities and hence decreases
the required computational time significantly. Comparison of Runge-Kutta and
ADER at their respective CFL stability limit renders ADER especially beneficial
for higher orders when the Butcher barrier implies an overproportional amount
of stages. Moreover, vector updates in explicit Runge--Kutta schemes are shown
to take a substantial amount of the computational time due to their memory
intensity
A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows
Both compressible and incompressible Navier-Stokes solvers can be used and
are used to solve incompressible turbulent flow problems. In the compressible
case, the Mach number is then considered as a solver parameter that is set to a
small value, , in order to mimic incompressible flows.
This strategy is widely used for high-order discontinuous Galerkin
discretizations of the compressible Navier-Stokes equations. The present work
raises the question regarding the computational efficiency of compressible DG
solvers as compared to a genuinely incompressible formulation. Our
contributions to the state-of-the-art are twofold: Firstly, we present a
high-performance discontinuous Galerkin solver for the compressible
Navier-Stokes equations based on a highly efficient matrix-free implementation
that targets modern cache-based multicore architectures. The performance
results presented in this work focus on the node-level performance and our
results suggest that there is great potential for further performance
improvements for current state-of-the-art discontinuous Galerkin
implementations of the compressible Navier-Stokes equations. Secondly, this
compressible Navier-Stokes solver is put into perspective by comparing it to an
incompressible DG solver that uses the same matrix-free implementation. We
discuss algorithmic differences between both solution strategies and present an
in-depth numerical investigation of the performance. The considered benchmark
test cases are the three-dimensional Taylor-Green vortex problem as a
representative of transitional flows and the turbulent channel flow problem as
a representative of wall-bounded turbulent flows