9,228 research outputs found
An explicit kernel-split panel-based Nystr\"om scheme for integral equations on axially symmetric surfaces
A high-order accurate, explicit kernel-split, panel-based, Fourier-Nystr\"om
discretization scheme is developed for integral equations associated with the
Helmholtz equation in axially symmetric domains. Extensive incorporation of
analytic information about singular integral kernels and on-the-fly computation
of nearly singular quadrature rules allow for very high achievable accuracy,
also in the evaluation of fields close to the boundary of the computational
domain.Comment: 30 pages, 5 figures, errata correcte
Multilevel Sparse Grid Methods for Elliptic Partial Differential Equations with Random Coefficients
Stochastic sampling methods are arguably the most direct and least intrusive
means of incorporating parametric uncertainty into numerical simulations of
partial differential equations with random inputs. However, to achieve an
overall error that is within a desired tolerance, a large number of sample
simulations may be required (to control the sampling error), each of which may
need to be run at high levels of spatial fidelity (to control the spatial
error). Multilevel sampling methods aim to achieve the same accuracy as
traditional sampling methods, but at a reduced computational cost, through the
use of a hierarchy of spatial discretization models. Multilevel algorithms
coordinate the number of samples needed at each discretization level by
minimizing the computational cost, subject to a given error tolerance. They can
be applied to a variety of sampling schemes, exploit nesting when available,
can be implemented in parallel and can be used to inform adaptive spatial
refinement strategies. We extend the multilevel sampling algorithm to sparse
grid stochastic collocation methods, discuss its numerical implementation and
demonstrate its efficiency both theoretically and by means of numerical
examples
On discretely entropy conservative and entropy stable discontinuous Galerkin methods
High order methods based on diagonal-norm summation by parts operators can be
shown to satisfy a discrete conservation or dissipation of entropy for
nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as
nodal discontinuous Galerkin methods with diagonal mass matrices. In this work,
we describe how use flux differencing, quadrature-based projections, and
SBP-like operators to construct discretely entropy conservative schemes for DG
methods under more arbitrary choices of volume and surface quadrature rules.
The resulting methods are semi-discretely entropy conservative or entropy
stable with respect to the volume quadrature rule used. Numerical experiments
confirm the stability and high order accuracy of the proposed methods for the
compressible Euler equations in one and two dimensions
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