2,124 research outputs found
Pressure-induced diamond to beta-tin transition in bulk silicon: a near-exact quantum Monte Carlo study
The pressure-induced structural phase transition from diamond to beta-tin in
silicon is an excellent test for theoretical total energy methods. The
transition pressure provides a sensitive measure of small relative energy
changes between the two phases (one a semiconductor and the other a semimetal).
Experimentally, the transition pressure is well characterized.
Density-functional results have been unsatisfactory. Even the generally much
more accurate diffusion Monte Carlo method has shown a noticeable fixed-node
error. We use the recently developed phaseless auxiliary-field quantum Monte
Carlo (AFQMC) method to calculate the relative energy differences in the two
phases. In this method, all but the error due to the phaseless constraint can
be controlled systematically and driven to zero. In both structural phases we
were able to benchmark the error of the phaseless constraint by carrying out
exact unconstrained AFQMC calculations for small supercells. Comparison between
the two shows that the systematic error in the absolute total energies due to
the phaseless constraint is well within 0.5 mHa/atom. Consistent with these
internal benchmarks, the transition pressure obtained by the phaseless AFQMC
from large supercells is in very good agreement with experiment.Comment: 9 pages, 5 figure
On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations
The present paper deals with the numerical solution of the incompressible
Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods
for discretization in space. For DG methods applied to the dual splitting
projection method, instabilities have recently been reported that occur for
coarse spatial resolutions and small time step sizes. By means of numerical
investigation we give evidence that these instabilities are related to the
discontinuous Galerkin formulation of the velocity divergence term and the
pressure gradient term that couple velocity and pressure. Integration by parts
of these terms with a suitable definition of boundary conditions is required in
order to obtain a stable and robust method. Since the intermediate velocity
field does not fulfill the boundary conditions prescribed for the velocity, a
consistent boundary condition is derived from the convective step of the dual
splitting scheme to ensure high-order accuracy with respect to the temporal
discretization. This new formulation is stable in the limit of small time steps
for both equal-order and mixed-order polynomial approximations. Although the
dual splitting scheme itself includes inf-sup stabilizing contributions, we
demonstrate that spurious pressure oscillations appear for equal-order
polynomials and small time steps highlighting the necessity to consider inf-sup
stability explicitly.Comment: 31 page
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