257,288 research outputs found
Improving operational reliability of small hydro power plants in united energy system of Ukraine
We make an initial investigation into the temporal efficiency of a fully discrete summation-by-parts approach for unsteady flows. As a model problem for the Navier–Stokes equations we consider a two-dimensional advection–diffusion problem with a boundary layer. The problem is discretized in space using finite difference approximations on summation-by-parts form together with weak boundary conditions, leading to optimal stability estimates. For the time integration part we consider various forms of high order summation-by-parts operators and compare with an existing popular fourth order diagonally implicit Runge–Kutta method. To solve the resulting fully discrete equation system, we employ a multi-grid scheme with dual time stepping
Journal Staff
A time-dependent coordinate transformation of a constant coeffcient hyperbolic equation which results in a variable coeffcient problem is considered. By using the energy method, we derive well-posed boundary conditions for the continuous problem. It is shown that the number of boundary conditions depend on the coordinate transformation. By using Summation-by-Parts (SBP) operators for the space discretization and weak boundary conditions, an energy stable finite dieffrence scheme is obtained. We also show how to construct a time-dependent penalty formulation that automatically imposes the right number of boundary conditions. Numerical calculations corroborate the stability and accuracy of the approximations
COFFEE -- An MPI-parallelized Python package for the numerical evolution of differential equations
COFFEE (ConFormal Field Equation Evolver) is a Python package primarily
developed to numerically evolve systems of partial differential equations over
time using the method of lines. It includes a variety of time integrators and
finite differencing stencils with the summation-by-parts property, as well as
pseudo-spectral functionality for angular derivatives of spin-weighted
functions. Some additional capabilities include being MPI-parallelisable on a
variety of different geometries, HDF data output and post processing scripts to
visualize data, and an actions class that allows users to create code for
analysis after each timestep.Comment: 12 pages, 1 figure, accepted to be published in Software
A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation
We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard
equation that satisfies the summation-by-parts simultaneous-approximation-term
(SBP-SAT) property. The latter permits us to show that the discrete free-energy
is bounded, and as a result, the scheme is provably stable. The scheme and the
stability proof are presented for general curvilinear three-dimensional
hexahedral meshes. We use the Bassi-Rebay 1 (BR1) scheme to compute interface
fluxes, and an IMplicit-EXplicit (IMEX) scheme to integrate in time. Lastly, we
test the theoretical findings numerically and present examples for two and
three-dimensional problems
Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes
We present a systematic method for constructing boundary conditions (numerical and physical) of the required accuracy, for compact (Pade-like) high-order finite-difference schemes for hyperbolic systems. First, a roper summation-by-parts formula is found for the approximate derivative. A 'simultaneous approximation term' (SAT) is then introduced to treat the boundary conditions. This procedure leads to time-stable schemes even in the system case. An explicit construction of the fourth-order compact case is given. Numerical studies are presented to verify the efficacy of the approach
Trace preserving quantum dynamics using a novel reparametrization-neutral summation-by-parts difference operator
We develop a novel numerical scheme for the simulation of dissipative quantum
dynamics following from two-body Lindblad master equations. All defining
continuum properties of the Lindblad dynamics, hermiticity, positivity and in
particular trace conservation of the evolved density matrix are preserved. The
central ingredient is a new spatial difference operator, which not only fulfils
the summation by parts (SBP) property but also implements a continuum
reparametrization property. Using the time evolution of a heavy-quark
anti-quark bound state in a hot thermal medium as an explicit example, we show
how the reparametrization neutral summation-by-parts (RN-SBP) operator
preserves the continuum properties of the theory.Comment: 34 pages, 7 figures, open-access code available via
https://doi.org/10.5281/zenodo.374446
High-order cyclo-difference techniques: An alternative to finite differences
The summation-by-parts energy norm is used to establish a new class of high-order finite-difference techniques referred to here as 'cyclo-difference' techniques. These techniques are constructed cyclically from stable subelements, and require no numerical boundary conditions; when coupled with the simultaneous approximation term (SAT) boundary treatment, they are time asymptotically stable for an arbitrary hyperbolic system. These techniques are similar to spectral element techniques and are ideally suited for parallel implementation, but do not require special collocation points or orthogonal basis functions. The principal focus is on methods of sixth-order formal accuracy or less; however, these methods could be extended in principle to any arbitrary order of accuracy
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