1,794 research outputs found
Notes on the Discontinuous Galerkin methods for the numerical simulation of hyperbolic equations 1 General Context 1.1 Bibliography
The roots of Discontinuous Galerkin (DG) methods is usually attributed to
Reed and Hills in a paper published in 1973 on the numerical approximation of
the neutron transport equation [18]. In fact, the adventure really started with
a rather thoroughfull series of five papers by Cockburn and Shu in the late
80's [7, 5, 9, 6, 8]. Then, the fame of the method, which could be seen as a
compromise between Finite Elements (the center of the method being a weak
formulation) and Finite Volumes (the basis functions are defined cell-wise, the
cells being the elements of the primal mesh) increased and slowly investigated
successfully all the domains of Partial Differential Equations numerical
integration. In particular, one can cite the ground papers for the common
treatment of convection-diffusion equations [4, 3] or the treatment of pure
elliptic equations [2, 17]. For more information on the history of
Discontinuous Galerkin method, please refer to section 1.1 of [15]. Today, DG
methods are widely used in all kind of manners and have applications in almost
all fields of applied mathematics. (TODO: cite applications and
structured/unstructured meshes, steady/unsteady, etc...). The methods is now
mature enough to deserve entire text books, among which I cite a reference book
on Nodal DG Methods by Henthaven and Warburton [15] with the ground basis of DG
integration, numerical analysis of its linear behavior and generalization to
multiple dimensions. Lately, since 2010, thanks to a ground work of Zhang and
Shu [26, 27, 25, 28, 29], Discontinuous Galerkin methods are eventually able to
combine high order accuracy and certain preservation of convex constraints,
such as the positivity of a given quantity, for example. These new steps
forward are very promising since it brings us very close to the "Ultimate
Conservative Scheme", [23, 1]
Reproducibility, accuracy and performance of the Feltor code and library on parallel computer architectures
Feltor is a modular and free scientific software package. It allows
developing platform independent code that runs on a variety of parallel
computer architectures ranging from laptop CPUs to multi-GPU distributed memory
systems. Feltor consists of both a numerical library and a collection of
application codes built on top of the library. Its main target are two- and
three-dimensional drift- and gyro-fluid simulations with discontinuous Galerkin
methods as the main numerical discretization technique. We observe that
numerical simulations of a recently developed gyro-fluid model produce
non-deterministic results in parallel computations. First, we show how we
restore accuracy and bitwise reproducibility algorithmically and
programmatically. In particular, we adopt an implementation of the exactly
rounded dot product based on long accumulators, which avoids accuracy losses
especially in parallel applications. However, reproducibility and accuracy
alone fail to indicate correct simulation behaviour. In fact, in the physical
model slightly different initial conditions lead to vastly different end
states. This behaviour translates to its numerical representation. Pointwise
convergence, even in principle, becomes impossible for long simulation times.
In a second part, we explore important performance tuning considerations. We
identify latency and memory bandwidth as the main performance indicators of our
routines. Based on these, we propose a parallel performance model that predicts
the execution time of algorithms implemented in Feltor and test our model on a
selection of parallel hardware architectures. We are able to predict the
execution time with a relative error of less than 25% for problem sizes between
0.1 and 1000 MB. Finally, we find that the product of latency and bandwidth
gives a minimum array size per compute node to achieve a scaling efficiency
above 50% (both strong and weak)
ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement
We present the first high order one-step ADER-WENO finite volume scheme with
Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial
accuracy is obtained through a WENO reconstruction, while a high order one-step
time discretization is achieved using a local space-time discontinuous Galerkin
predictor method. Due to the one-step nature of the underlying scheme, the
resulting algorithm is particularly well suited for an AMR strategy on
space-time adaptive meshes, i.e.with time-accurate local time stepping. The AMR
property has been implemented 'cell-by-cell', with a standard tree-type
algorithm, while the scheme has been parallelized via the Message Passing
Interface (MPI) paradigm. The new scheme has been tested over a wide range of
examples for nonlinear systems of hyperbolic conservation laws, including the
classical Euler equations of compressible gas dynamics and the equations of
magnetohydrodynamics (MHD). High order in space and time have been confirmed
via a numerical convergence study and a detailed analysis of the computational
speed-up with respect to highly refined uniform meshes is also presented. We
also show test problems where the presented high order AMR scheme behaves
clearly better than traditional second order AMR methods. The proposed scheme
that combines for the first time high order ADER methods with space--time
adaptive grids in two and three space dimensions is likely to become a useful
tool in several fields of computational physics, applied mathematics and
mechanics.Comment: With updated bibliography informatio
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
Multigrid Preconditioning for a Space-Time Spectral-Element Discontinuous-Galerkin Solver
In this work we examine a multigrid preconditioning approach in the context of a high- order tensor-product discontinuous-Galerkin spectral-element solver. We couple multigrid ideas together with memory lean and efficient tensor-product preconditioned matrix-free smoothers. Block ILU(0)-preconditioned GMRES smoothers are employed on the coarsest spaces. The performance is evaluated on nonlinear problems arising from unsteady scale- resolving solutions of the Navier-Stokes equations: separated low-Mach unsteady ow over an airfoil from laminar to turbulent ow. A reduction in the number of ne space iterations is observed, which proves the efficiency of the approach in terms of preconditioning the linear systems, however this gain was not reflected in the CPU time. Finally, the preconditioner is successfully applied to problems characterized by stiff source terms such as the set of RANS equations, where the simple tensor product preconditioner fails. Theoretical justification about the findings is reported and future work is outlined
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