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21
Energy conserving explicit local timestepping for secondorder wave equations
 SIAM J. Sci. Comput
"... Abstract. Locally refined meshes impose severe stability constraints on explicit timestepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local timestepping methods are developed, which allow arbitrarily small time steps precisely ..."
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Cited by 20 (3 self)
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Abstract. Locally refined meshes impose severe stability constraints on explicit timestepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local timestepping methods are developed, which allow arbitrarily small time steps precisely where small elements in the mesh are located. When combined with a symmetric finite element discretization in space with an essentially diagonal mass matrix, the resulting discrete numerical scheme is explicit, is inherently parallel, and exactly conserves a discrete energy. Starting from the standard secondorder “leapfrog ” scheme, timestepping methods of arbitrary order of accuracy are derived. Numerical experiments illustrate the efficiency and usefulness of these methods and validate the theory. Key words. secondorder hyperbolic problems, explicit methods, time reversible methods, energy conservation, discontinuous Galerkin methods, finite element methods, mass lumping, wave equation, acoustic waves, electromagnetic waves
Difference approximations of the Neumann problem for the second order wave equation
 SIAM J. Numer. Anal
, 2004
"... Abstract. Stability theory and numerical experiments are presented for a finite difference method that directly discretizes the Neumann problem for the second order wave equation. Complex geometries are discretized using a Cartesian embedded boundary technique. Both second and third order accurate a ..."
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Cited by 9 (1 self)
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Abstract. Stability theory and numerical experiments are presented for a finite difference method that directly discretizes the Neumann problem for the second order wave equation. Complex geometries are discretized using a Cartesian embedded boundary technique. Both second and third order accurate approximations of the boundary conditions are presented. Away from the boundary, the basic second order method can be corrected to achieve fourth order spatial accuracy. To integrate in time, we present both a second order and a fourth order accurate explicit method. The stability of the method is ensured by adding a small fourth order dissipation operator, locally modified near the boundary to allow its application at all grid points inside the computational domain. Numerical experiments demonstrate the accuracy and longtime stability of the proposed method.
A Wellposed and discretely stable perfectly matched layer for elastic wave equations in second order formulation
, 2010
"... Abstract. We present a wellposed and discretely stable perfectly matched layer for the anisotropic (and isotropic) elastic wave equations without first rewriting the governing equations as a first order system. The new model is derived by the complex coordinate stretching technique. Using standard ..."
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Cited by 5 (2 self)
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Abstract. We present a wellposed and discretely stable perfectly matched layer for the anisotropic (and isotropic) elastic wave equations without first rewriting the governing equations as a first order system. The new model is derived by the complex coordinate stretching technique. Using standard perturbation methods we show that complex frequency shift together with a chosen real scaling factor ensures the decay of eigenmodes for all relevant frequencies. To buttress the stability properties and the robustness of the proposed model, numerical experiments are presented for anisotropic elastic wave equations. The model is approximated with a stable nodecentered finite difference scheme that is second order accurate both in time and space.
High order marching schemes for the wave equation in complex geometry
 J. Comput. Phys
, 2004
"... Abstract We present a new class of explicit marching schemes for the wave equation in complex geometry. They rely on a simple embedding of the domain in a uniform Cartesian grid, which allows for efficient and automatic implementation but creates irregular cells near the boundary. While existing ex ..."
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Cited by 3 (0 self)
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Abstract We present a new class of explicit marching schemes for the wave equation in complex geometry. They rely on a simple embedding of the domain in a uniform Cartesian grid, which allows for efficient and automatic implementation but creates irregular cells near the boundary. While existing explicit finite difference schemes are generally restricted in the size of the time step that can be taken by the dimensions of the smallest cell, the schemes described here are capable of taking time steps dictated by the uniform grid spacing. This should be of significant benefit in a wide variety of simulation efforts.
An embedded boundary method for the wave equation with discontinuous coefficients
 SIAM J. Sci. Comput
"... Abstract A second order accurate embedded boundary method for the twodimensional wave equation with discontinuous wave propagation speed is described. The wave equation is discretized on a Cartesian grid with constant grid size and the interface (across which the wave speed is discontinuous) is al ..."
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Cited by 2 (0 self)
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Abstract A second order accurate embedded boundary method for the twodimensional wave equation with discontinuous wave propagation speed is described. The wave equation is discretized on a Cartesian grid with constant grid size and the interface (across which the wave speed is discontinuous) is allowed to intersect the mesh in an arbitrary fashion. By using ghost points on either side of the interface, previous embedded boundary techniques for the Neumann and Dirichlet problems are generalized to satisfy the jump conditions across the interface to second order accuracy. The resulting discretization of the jump conditions has the desirable property that each ghost point can be updated independently of all other ghost points, resulting in a fully explicit timeintegration method. Numerical examples are given where the method is used to study electromagnetic scattering of a plane wave by a dielectric cylinder. The numerical solutions are evaluated against the analytical solution due to Mie, and pointwise second order accuracy is confirmed.
Analysis of wave propagation in 1D inhomogeneous media
 Num. Funct. Anal. Opt
"... In this paper we consider the one dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In a first part of the paper we analyze the asymptotic nodal point distribution of high frequency eigenfunctions, which, in turn gives f ..."
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In this paper we consider the one dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In a first part of the paper we analyze the asymptotic nodal point distribution of high frequency eigenfunctions, which, in turn gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the high and low frequency limit. In the latter case we derive an homogenization limit whereas in the first we show that a sort of selfhomogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequencydependent measure. The proposed scheme yields accurate resolution of both high and low frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and nonsmooth speeds of propagation.
Perfectly Matched Layers for Second Order Wave Equations
, 2010
"... Numerical simulation of propagating waves in unbounded spatial domains is a challenge common to many branches of engineering and applied mathematics. Perfectly matched layers (PML) are a novel technique for simulating the absorption of waves in open domains. The equations modeling the dynamics of ph ..."
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Cited by 1 (1 self)
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Numerical simulation of propagating waves in unbounded spatial domains is a challenge common to many branches of engineering and applied mathematics. Perfectly matched layers (PML) are a novel technique for simulating the absorption of waves in open domains. The equations modeling the dynamics of phenomena of interest are usually posed as differential equations (or integral equations) which must be solved at every time instant. In many application areas like general relativity, seismology and acoustics, the underlying equations are systems of second order hyperbolic partial differential equations. In numerical treatment of such problems, the equations are often rewritten as first order systems and are solved in this form. For this reason, many existing PML models have been developed for first order systems. In several studies, it has been reported that there are drawbacks with rewriting second order systems into first order systems before numerical solutions are obtained. While the theory and numerical methods for first order systems are well developed, numerical techniques to solve second order hyperbolic
RBID08223Z) “Advanced numerical techniques for uncertainty quantification in engineering and life
"... A stochastic collocation method for the second order wave equation with a discontinuous random speed ..."
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A stochastic collocation method for the second order wave equation with a discontinuous random speed
A New Skew Linear Interpolation Characteristic Difference Method for Sobolev Equation
, 2011
"... Abstract. A new kind of characteristicdifference scheme for Sobolev equations is constructed by combining characteristic method with the finitedifference method and with the skew linear interpolation method. The convergence of the characteristicdifference scheme is studied. The advantage of this ..."
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Abstract. A new kind of characteristicdifference scheme for Sobolev equations is constructed by combining characteristic method with the finitedifference method and with the skew linear interpolation method. The convergence of the characteristicdifference scheme is studied. The advantage of this scheme is very effectual to eliminate the numerical oscillations and have potential advantages in other equations.
with random coefficients
, 2012
"... Analysis and computation of the elastic wave equation with random coefficients ..."
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Analysis and computation of the elastic wave equation with random coefficients