An important step in estimating the index of refraction of electromagnetic
scattering problems is to compute the associated transmission eigenvalue problem.
We develop in this paper efficient and accurate spectral methods for computing the
transmission eigenvalues associated to the electromagnetic scattering problems. We
present ample numerical results to show that our methods are very effective for computing transmission eigenvalues (particularly for computing the smallest eigenvalue),
and together with the linear sampling method, provide an efficient way to estimate the
index of refraction of a non-absorbing inhomogeneous medium.
In this paper, we propose two hexagonal Fourier-Galerkin methods for the
direct numerical simulation of the two-dimensional homogeneous isotropic decaying
turbulence. We first establish the lattice Fourier analysis as a mathematical foundation. Then a universal approximation scheme is devised for our hexagonal Fourier-Galerkin methods for Navier-Stokes equations. Numerical experiments mainly concentrate on the decaying properties and the self-similar spectra of the two-dimensional
homogeneous turbulence at various initial Reynolds numbers with an initial flow field
governed by a Gaussian-distributed energy spectrum. Numerical results demonstrate
that both the hexagonal Fourier-Galerkin methods are as efficient as the classic square
Fourier-Galerkin method, while provide more effective statistical physical quantities
In this paper, we devote ourselves to the research of numerical methods
for American option pricing problems under the Black-Scholes model. The optimal
exercise boundary which satisfies a nonlinear Volterra integral equation is resolved by
a high-order collocation method based on graded meshes. For the other spatial domain
boundary, an artificial boundary condition is applied to the pricing problem for the
effective truncation of the semi-infinite domain. Then, the front-fixing and stretching
transformations are employed to change the truncated problem in an irregular domain
into a one-dimensional parabolic problem in [−1,1]. The Chebyshev spectral method
coupled with fourth-order Runge-Kutta method is proposed for the resulting parabolic
problem related to the options. The stability of the semi-discrete numerical method is
established for the parabolic problem transformed from the original model. Numerical
experiments are conducted to verify the performance of the proposed methods and
compare them with some existing methods.
This paper is devoted to stability analysis of the acoustic wave equation
exterior to a bounded scatterer, where the unbounded computational domain is truncated by the exact time-domain circular/spherical nonreflecting boundary condition
(NRBC). Different from the usual energy method, we adopt an argument that leads
to $L^2$-a priori estimates with minimum regularity requirement for the initial data and
source term. This needs some delicate analysis of the involved NRBC. These results
play an essential role in the error analysis of the interior solvers (e.g., finite-element/spectral-
element/spectral methods) for the reduced scattering problems. We also apply the
technique to analyze a time-domain waveguide problem.
In this paper, we carry out an a posteriori error analysis of Legendre spectral
approximations to the Stokes/Darcy coupled equations. The spectral approximations
are based on a weak formulation of the coupled equations by using the Beavers-Joseph-Saffman interface condition. The main contribution of the paper consists of deriving
a number of posteriori error indicators and their upper and lower bounds for the single domain case. An extension of the upper bounds to the multi-domain case in the
spectral element framework is also given.
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