38,937 research outputs found
An optimal polynomial approximation of Brownian motion
In this paper, we will present a strong (or pathwise) approximation of
standard Brownian motion by a class of orthogonal polynomials. The coefficients
that are obtained from the expansion of Brownian motion in this polynomial
basis are independent Gaussian random variables. Therefore it is practical
(requires independent Gaussian coefficients) to generate an approximate
sample path of Brownian motion that respects integration of polynomials with
degree less than . Moreover, since these orthogonal polynomials appear
naturally as eigenfunctions of an integral operator defined by the Brownian
bridge covariance function, the proposed approximation is optimal in a certain
weighted sense. In addition, discretizing Brownian paths as
piecewise parabolas gives a locally higher order numerical method for
stochastic differential equations (SDEs) when compared to the standard
piecewise linear approach. We shall demonstrate these ideas by simulating
Inhomogeneous Geometric Brownian Motion (IGBM). This numerical example will
also illustrate the deficiencies of the piecewise parabola approximation when
compared to a new version of the asymptotically efficient log-ODE (or
Castell-Gaines) method.Comment: 27 pages, 8 figure
Improved approximation of phase-space densities on triangulated domains using Discrete Flow Mapping with p-refinement
We consider the approximation of the phase-space flow of a dynamical system on a triangulated surface using an approach known as Discrete Flow Mapping. Such flows are of interest throughout statistical mechanics, but the focus here is on flows arising from ray tracing approximations of linear wave equations. An orthogonal polynomial basis approximation of the phase-space density is applied in both the position and direction coordinates, in contrast with previous studies where piecewise constant functions have typically been applied for the spatial approximation. In order to improve the tractability of an orthogonal polynomial approximation in both phase-space coordinates, we propose a careful strategy for computing the propagation operator. For the favourable case of a Legendre polynomial basis we show that the integrals in the definition of the propagation operator may be evaluated analytically with respect to position and via a spectrally convergent quadrature rule for the direction coordinate. A generally applicable spectral quadrature scheme for integration with respect to both coordinates is also detailed for completeness. Finally, we provide numerical results that motivate the use of p-refinement in the orthogonal polynomial basis
High-order conservative reconstruction schemes for finite volume methods in cylindrical and spherical coordinates
High-order reconstruction schemes for the solution of hyperbolic conservation
laws in orthogonal curvilinear coordinates are revised in the finite volume
approach. The formulation employs a piecewise polynomial approximation to the
zone-average values to reconstruct left and right interface states from within
a computational zone to arbitrary order of accuracy by inverting a
Vandermonde-like linear system of equations with spatially varying
coefficients. The approach is general and can be used on uniform and
non-uniform meshes although explicit expressions are derived for polynomials
from second to fifth degree in cylindrical and spherical geometries with
uniform grid spacing. It is shown that, in regions of large curvature, the
resulting expressions differ considerably from their Cartesian counterparts and
that the lack of such corrections can severely degrade the accuracy of the
solution close to the coordinate origin. Limiting techniques and monotonicity
constraints are revised for conventional reconstruction schemes, namely, the
piecewise linear method (PLM), third-order weighted essentially non-oscillatory
(WENO) scheme and the piecewise parabolic method (PPM).
The performance of the improved reconstruction schemes is investigated in a
number of selected numerical benchmarks involving the solution of both scalar
and systems of nonlinear equations (such as the equations of gas dynamics and
magnetohydrodynamics) in cylindrical and spherical geometries in one and two
dimensions. Results confirm that the proposed approach yields considerably
smaller errors, higher convergence rates and it avoid spurious numerical
effects at a symmetry axis.Comment: 37 pages, 12 Figures. Accepted for publication in Journal of
Compuational Physic
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