9,798 research outputs found
Object-oriented construction of a multigrid electronic-structure code with Fortran 90
We describe the object-oriented implementation of a higher-order
finite-difference density-functional code in Fortran 90. Object-oriented models
of grid and related objects are constructed and employed for the implementation
of an efficient one-way multigrid method we have recently proposed for the
density-functional electronic-structure calculations. Detailed analysis of
performance and strategy of the one-way multigrid scheme will be presented.Comment: 24 pages, 6 figures, to appear in Comput. Phys. Com
Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations
We analyse two practical aspects that arise in the numerical solution of
Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone
approximation schemes known as semi-Lagrangian schemes. These schemes make use
of a wide stencil to achieve convergence and result in discretization matrices
that are less sparse and less local than those coming from standard finite
difference schemes. This leads to computational difficulties not encountered
there. In particular, we consider the overstepping of the domain boundary and
analyse the accuracy and stability of stencil truncation. This truncation
imposes a stricter CFL condition for explicit schemes in the vicinity of
boundaries than in the interior, such that implicit schemes become attractive.
We then study the use of geometric, algebraic and aggregation-based multigrid
preconditioners to solve the resulting discretised systems from implicit time
stepping schemes efficiently. Finally, we illustrate the performance of these
techniques numerically for benchmark test cases from the literature
Multigrid optimization for space-time discontinuous Galerkin discretizations of advection dominated flows
The goal of this research is to optimize multigrid methods for higher order accurate space-time discontinuous Galerkin discretizations. The main analysis tool is discrete Fourier analysis of two- and three-level multigrid algorithms. This gives the spectral radius of the error transformation operator which predicts the asymptotic rate of convergence of the multigrid algorithm. In the optimization process we therefore choose to minimize the spectral radius of the error transformation operator. We specifically consider optimizing h-multigrid methods with explicit Runge-Kutta type smoothers for second and third order accurate space-time discontinuous Galerkin finite element discretizations of the 2D advection-diffusion equation. The optimized schemes are compared with current h-multigrid techniques employing Runge-Kutta type smoothers. Also, the efficiency of h-, p- and hp-multigrid methods for solving the Euler equations of gas dynamics with a higher order accurate space-time DG method is investigated
A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory
We develop a new dispersion minimizing compact finite difference scheme for
the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly
developed ray theory for difference equations. A discrete Helmholtz operator
and a discrete operator to be applied to the source and the wavefields are
constructed. Their coefficients are piecewise polynomial functions of ,
chosen such that phase and amplitude errors are minimal. The phase errors of
the scheme are very small, approximately as small as those of the 2-D
quasi-stabilized FEM method and substantially smaller than those of
alternatives in 3-D, assuming the same number of gridpoints per wavelength is
used. In numerical experiments, accurate solutions are obtained in constant and
smoothly varying media using meshes with only five to six points per wavelength
and wave propagation over hundreds of wavelengths. When used as a coarse level
discretization in a multigrid method the scheme can even be used with downto
three points per wavelength. Tests on 3-D examples with up to degrees of
freedom show that with a recently developed hybrid solver, the use of coarser
meshes can lead to corresponding savings in computation time, resulting in good
simulation times compared to the literature.Comment: 33 pages, 12 figures, 6 table
One-way multigrid method in electronic structure calculations
We propose a simple and efficient one-way multigrid method for
self-consistent electronic structure calculations based on iterative
diagonalization. Total energy calculations are performed on several different
levels of grids starting from the coarsest grid, with wave functions
transferred to each finer level. The only changes compared to a single grid
calculation are interpolation and orthonormalization steps outside the original
total energy calculation and required only for transferring between grids. This
feature results in a minimal amount of code change, and enables us to employ a
sophisticated interpolation method and noninteger ratio of grid spacings.
Calculations employing a preconditioned conjugate gradient method are presented
for two examples, a quantum dot and a charged molecular system. Use of three
grid levels with grid spacings 2h, 1.5h, and h decreases the computer time by
about a factor of 5 compared to single level calculations.Comment: 10 pages, 2 figures, to appear in Phys. Rev. B, Rapid Communication
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