22,358 research outputs found
Hyperboloidal layers for hyperbolic equations on unbounded domains
We show how to solve hyperbolic equations numerically on unbounded domains by
compactification, thereby avoiding the introduction of an artificial outer
boundary. The essential ingredient is a suitable transformation of the time
coordinate in combination with spatial compactification. We construct a new
layer method based on this idea, called the hyperboloidal layer. The method is
demonstrated on numerical tests including the one dimensional Maxwell equations
using finite differences and the three dimensional wave equation with and
without nonlinear source terms using spectral techniques.Comment: 23 pages, 23 figure
Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisations of the Maxwell equations
Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed. A comparison between the most popular choices of Runge-Kutta (RK) methods is made from the point of view of accuracy and computational work. By choosing the strong-stability-preserving Runge-Kutta (SSP-RK) time-integration method of order consistent with the polynomial order of the spatial discretisation, better accuracy can be attained compared with fixed-order schemes. Moreover, this comes without a significant increase in the computational work. A numerical Fourier analysis is performed for this Runge-Kutta discontinuous Galerkin (RKDG) discretisation to gain insight into the dispersion and dissipation properties of the fully discrete scheme. The analysis is carried out on both the one-dimensional and the two-dimensional fully discrete schemes and, in the latter case, on uniform as well as on non-uniform meshes. It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme
Solving seismic wave propagation in elastic media using the matrix exponential approach
Three numerical algorithms are proposed to solve the time-dependent
elastodynamic equations in elastic solids. All algorithms are based on
approximating the solution of the equations, which can be written as a matrix
exponential. By approximating the matrix exponential with a product formula, an
unconditionally stable algorithm is derived that conserves the total elastic
energy density. By expanding the matrix exponential in Chebyshev polynomials
for a specific time instance, a so-called ``one-step'' algorithm is constructed
that is very accurate with respect to the time integration. By formulating the
conventional velocity-stress finite-difference time-domain algorithm (VS-FDTD)
in matrix exponential form, the staggered-in-time nature can be removed by a
small modification, and higher order in time algorithms can be easily derived.
For two different seismic events the accuracy of the algorithms is studied and
compared with the result obtained by using the conventional VS-FDTD algorithm.Comment: 13 pages revtex, 6 figures, 2 table
Conservative and non-conservative methods based on hermite weighted essentially-non-oscillatory reconstruction for Vlasov equations
We introduce a WENO reconstruction based on Hermite interpolation both for
semi-Lagrangian and finite difference methods. This WENO reconstruction
technique allows to control spurious oscillations. We develop third and fifth
order methods and apply them to non-conservative semi-Lagrangian schemes and
conservative finite difference methods. Our numerical results will be compared
to the usual semi-Lagrangian method with cubic spline reconstruction and the
classical fifth order WENO finite difference scheme. These reconstructions are
observed to be less dissipative than the usual weighted essentially non-
oscillatory procedure. We apply these methods to transport equations in the
context of plasma physics and the numerical simulation of turbulence phenomena
Exponential Krylov time integration for modeling multi-frequency optical response with monochromatic sources
Light incident on a layer of scattering material such as a piece of sugar or
white paper forms a characteristic speckle pattern in transmission and
reflection. The information hidden in the correlations of the speckle pattern
with varying frequency, polarization and angle of the incident light can be
exploited for applications such as biomedical imaging and high-resolution
microscopy. Conventional computational models for multi-frequency optical
response involve multiple solution runs of Maxwell's equations with
monochromatic sources. Exponential Krylov subspace time solvers are promising
candidates for improving efficiency of such models, as single monochromatic
solution can be reused for the other frequencies without performing full
time-domain computations at each frequency. However, we show that the
straightforward implementation appears to have serious limitations. We further
propose alternative ways for efficient solution through Krylov subspace
methods. Our methods are based on two different splittings of the unknown
solution into different parts, each of which can be computed efficiently.
Experiments demonstrate a significant gain in computation time with respect to
the standard solvers.Comment: 22 pages, 4 figure
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