Time-dependent Wave Equation Research Articles

Optimal Error Estimates for the Fully Discrete Interior Penalty DG Method for the Wave Equation

In Grote et al. (SIAM J. Numer. Anal., 44:2408---2431, 2006) a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L 2-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation (leap-frog scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1+Δt 2), where p denotes the polynomial degree, h the mesh size, and Δt the time step.

Temporal behavior of dark spatial solitons in closed-circuit photovoltaic media

We show that the time-dependent nonlinear wave equation in closed-circuit photovoltaic media can exhibit quasi-steady-state and steady-state spatial solitons. We demonstrate that the formation time of open-circuit quasi-steady-state and open-circuit steady-state dark solitons decreases with an increase in the intensity ratio of the soliton, which is the ratio between the soliton peak intensity and the dark irradiance. We find that for the time-dependent nonlinear wave equation that exhibits only an open-circuit steady-state dark soliton, changing the electric current density J 0 does not generate quasi-steady-state dark solitons and affects the formation time of steady-state dark solitons and that for the time-dependent nonlinear wave equation that exhibits an open-circuit quasi-steady-state dark soliton, changing J 0 gives rise to three different time evolution regimes of the full width half maximum of the soliton’s intensity. The first regime shows that the formation time of steady-state dark solitons increases with J 0 whereas the formation time of quasi-steady-state dark solitons is independent of J 0. The second regime shows that the formation time of steady-state dark solitons decreases with an increases in J 0 and the formation time of quasi-steady-state dark solitons increases with J 0. The third regime shows that changing J 0 enables only steady-state dark solitons in the time-dependent nonlinear wave equation, of which the formation time increases with J 0.

High-order local absorbing conditions for the wave equation: Extensions and improvements

The solution of the time-dependent wave equation in an unbounded domain is considered. An artificial boundary B is introduced which encloses a finite computational domain. On B an absorbing boundary condition (ABC) is imposed. A formulation of local high-order ABCs recently proposed by Hagstrom and Warburton and based on a modification of the Higdon ABCs, is further developed and extended in a number of ways. First, the ABC is analyzed in new ways and important information is extracted from this analysis. Second, The ABCs are extended to the case of a dispersive medium, for which the Klein–Gordon wave equation governs. Third, the case of a stratified medium is considered and the way to apply the ABCs to this case is explained. Fourth, the ABCs are extended to take into account evanescent modes in the exact solution. The analysis is applied throughout this paper to two-dimensional wave guides. Two numerical algorithms incorporating these ABCs are considered: a standard semi-discrete finite element formulation in space followed by time-stepping, and a high-order finite difference discretization in space and time. Numerical examples are provided to demonstrate the performance of the extended ABCs using these two methods.

Acoustic Wave Propagator—A Split Region Implementation

In this article we develop and implement a split region technique that solves the time-dependent acoustic wave equation with greatly increased efficiency. This method uses a sophisticated Chebyshev propagation scheme in areas where there are interfaces and medium variations, and a simple free space propagator where the medium is homogeneous. Mathematica provides a cohesive and interactive environment, where the mathematical functions and visualization tools required for this work are already built in. The interactive interface allows users to modify the code and study specific problems with ease.

Emergent semiclassical time in quantum gravity: II. Full geometrodynamics and minisuperspace examples

I apply the preceding paper's emergent semiclassical time approach to geometrodynamics. The analogy between the two papers is useful at the level of the quadratic constraints, while I document the differences between the two due to the underlying differences in their linear constraints. I find that the emergent time-dependent wave equation for the universe in general not a time-dependent Schrödinger equation but rather a more general equation containing second time derivatives, and estimate in which regime this becomes significant. I provide a specific minisuperspace example for my emergent semiclassical time scheme and compare it with the hidden York time scheme. Overall, interesting connections are shown between Newtonian, Leibniz–Mach–Barbour, Wentzel–Kramers–Brillouin (WKB) and cosmic times, while the Euler and York hidden dilational times are argued to be somewhat different from these.

Controllability method for the Helmholtz equation with higher-order discretizations

We consider a controllability technique for the numerical solution of the Helmholtz equation. The original time-harmonic equation is represented as an exact controllability problem for the time-dependent wave equation. This problem is then formulated as a least-squares optimization problem, which is solved by the conjugate gradient method. Such an approach was first suggested and developed in the 1990s by French researchers and we introduce some improvements to its practical realization. We use higher-order spectral elements for spatial discretization, which leads to high accuracy and lumped mass matrices. Higher-order approximation reduces the pollution effect associated with finite element approximation of time-harmonic wave equations, and mass lumping makes explicit time-stepping schemes for the wave equation very efficient. We also derive a new way to compute the gradient of the least-squares functional and use algebraic multigrid method for preconditioning the conjugate gradient algorithm. Numerical results demonstrate the significant improvements in efficiency due to the higher-order spectral elements. For a given accuracy, spectral element method requires fewer computational operations than conventional finite element method. In addition, by using higher-order polynomial basis the influence of the pollution effect is reduced.

Controllability method for acoustic scattering with spectral elements

We formulate the Helmholtz equation as an exact controllability problem for the time-dependent wave equation. The problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements. This approach leads to high accuracy in spatial discretization. Moreover, the spectral element method results in diagonal mass matrices, which makes the time integration of the wave equation highly efficient. After discretization, the exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method. We illustrate the method with some numerical experiments, which demonstrate the significant improvements in efficiency due to the higher order spectral elements. For a given accuracy, the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method. In addition, by using higher order polynomial basis the influence of the pollution effect is reduced.

Kinetic effects on relativistic solitons in plasmas

We present theoretical and numerical studies of nonlinear interactions between a short, intense laser-beam and a plasma containing hot electrons. Such plasmas are frequently observed in fast-ignition experiments where electrons are heated by intense laser beams. Due to the combined effects of relativistic electron mass increase and relativistic ponderomotive force of the intense laser fields, there appear localized laser pulses in association with large amplitude localized positive potentials. The latter can trap and accelerate electrons to ultra-relativistic energies. This is demonstrated by means of computer simulations of the time-dependent nonlinear electromagnetic wave equation, supplemented by the Vlasov and Poisson equations.

Numerical Simulation of Q-Switched Supersonic Flow Chemical Oxygen-Iodine Laser by Solving Time-Dependent Paraxial Wave Equation

The quality and power of an extracted beam from a Q-switched supersonic flow chemical oxygen-iodine laser have been investigated by numerical simulation. The flow system adopted in this study is the throat-mixing system proposed in a previous paper. Three-dimensional calculation for optics is coupled with one-dimensional calculation for gas flow including chemical reactions. Both geometric and wave optics are calculated and compared to assess the effects of diffraction. Wave optics is calculated with a time-dependent paraxial wave equation. The results indicate that wave and geometric optics are qualitatively similar in the time-dependent behavior of beam power. Quantitatively, it is found that diffraction reduces the extracted power by 9%. It is also found from the spreading half-angle of the beam for wave optics that the beam quality at a maximum power is equivalent to that of a plane wave; however, it is lower than that of continuous extraction.

An Adaptive Hybrid FEM/FDM Method for an Inverse Scattering Problem in Scanning Acoustic Microscopy

Scanning acoustic microscopy based on focused ultrasound waves is a promising new tool in medical imaging. In this work we apply an adaptive hybrid FEM/FDM (finite element methods/finite difference methods) method to an inverse scattering problem for the time-dependent acoustic wave equation, where one seeks to reconstruct an unknown sound velocity c(x) from a single measurement of wave-reflection data on a small part of the boundary, e.g., to detect pathological defects in bone. Typically, this corresponds to identifying an unknown object (scatterer) in a surrounding homogeneous medium. The inverse problem is formulated as an optimal control problem, where we use an adjoint method to solve the equations of optimality expressing stationarity of an associated augmented Lagrangian by a quasi-Newton method. To treat the problem of multiple minima of the objective function, the optimization procedure is first performed on a coarse grid to smooth the high frequency error, generating a starting point for optimization steps on successively refined meshes. Local refinement based on the results of previous steps will improve computational efficiency of the method. As the main result then, an a posteriori error estimate is proved for the error in the Lagrangian, and a corresponding adaptive method is formulated, where the finite element mesh is refined from residual feedback. The performance of the adaptive hybrid method and the usefulness of the a posteriori error estimator for problems with limited boundary data are illustrated in three dimensional numerical examples.

High‐order Higdon‐like boundary conditions for exterior transient wave problems

AbstractRecently developed non‐reflecting boundary conditions are applied for exterior time‐dependent wave problems in unbounded domains. The linear time‐dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary ℬ︁, and a high‐order non‐reflecting boundary condition (NRBC) is imposed on ℬ︁. Then the problem is solved numerically in the finite domain bounded by ℬ︁. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high‐order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on ℬ︁. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two‐dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary‐variable formulation long‐time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high‐derivative formulation. Published in 2005 by John Wiley & Sons, Ltd.

Exact norm-conserving stochastic time-dependent Hartree–Fock

We derive an exact single-body decomposition of the time-dependent Schroedinger equation for N pairwise-interacting fermions. Each fermion obeys a stochastic time-dependent norm-preserving wave equation. As a first test of the method we calculate the low energy spectrum of Helium. An extension of the method to bosons is outlined.

Dirichlet-to-Neumann boundary condition for time-dependent dispersive waves in three-dimensional guides

Exact Dirichlet-to-Neumann (DtN) boundary conditions on cross-sections of three-dimensional semi-infinite wave guides are derived. This enables the exact truncation of a wave guide to allow computations in a finite domain Ω. The DtN boundary condition is nonlocal in space but local in time. Practical implementation requires the truncation of the exact boundary condition by approximating an infinite sum with a finite sum, and by terminating an infinite recursion relation. The truncated condition is incorporated in a finite element scheme to solve the problem in Ω. The cross-section of the guide may have an arbitrary shape. The three-dimensional time-dependent dispersive wave equation is considered in the guide. The dispersion parameter is allowed to vary in the cross-section. All the results reduce immediately to the two-dimensional and to the one-dimensional cases. To the best of the authors' knowledge, this is the first time that an exact boundary condition is derived in the dispersive case, even in one dimension.

Nonreflecting boundary conditions for the time-dependent convective wave equation in a duct

Solutions for the three-dimensional wave equation in an annular duct are represented in terms of Fourier–Bessel modes, each obeying a one-dimensional dispersive wave equation. An exact nonreflecting boundary condition, nonlocal in time but local in space, is derived for each mode. Since, in most applications, the number of propagating modes is a small finite subset of all the modes, the present condition yields a computationally efficient scheme. Convergence of the solution depends on the radial eigenvalues which characterize the dispersion of the propagating duct modes. For periodic forcing, convergence analysis shows that the solution tends toward its asymptotic limit as the reciprocal of the square root of time. Near a duct mode cut-on, the computational time required for convergence is proportional to the reciprocal of the square of the group velocity. The boundary condition is implemented numerically and tested by computing the propagation of dispersive waves with various frequencies and comparing the results with the analytic solution. Two local boundary conditions are also implemented and their performance relative to the nonlocal boundary condition is studied. Significant improvements in accuracy are observed by using the present boundary condition near cut-on. Numerical results also show that the rate of convergence of the solution to a time-periodic solution significantly decreases as the group velocity of the incident waves becomes small, consistent with the analytic results.

Exact boundary condition for time-dependent wave equation based on boundary integral

An exact non-reflecting boundary conditions based on a boundary integral equation or a modified Kirchhoff-type formula is derived for exterior three-dimensional wave equations. The Kirchhoff-type non-reflecting boundary condition is originally proposed by L. Ting and M.J. Miksis [J. Acoust. Soc. Am. 80 (1986) 1825] and numerically tested by D. Givoli and D. Cohen [J. Comput. Phys. 117 (1995) 102] for a spherically symmetric problem. The computational advantage of Ting–Miksis boundary condition is that its temporal non-locality is limited to a fixed amount of past information. However, a long-time instability is exhibited in testing numerical solutions by using a standard non-dissipative finite-difference scheme. The main purpose of this work is to present a new exact boundary condition and to eliminate the long-time instability. The proposed exact boundary condition can be considered as a limit case of Ting–Miksis boundary condition when the two artificial boundaries used in their method approach each other. Our boundary condition is actually a boundary integral equation on a single artificial boundary for wave equations, which is to be solved in conjunction with the interior wave equation. The new boundary condition needs only one artificial boundary, which can be of any shape, i.e., sphere, cubic surface, etc. It keeps all merits of the original Kirchhoff boundary condition such as restricting the temporal non-locality, free of numerical evaluation of any special functions and so on. Numerical approximation to the artificial boundary condition on cubic surface is derived and three-dimensional numerical tests are carried out on the cubic computational domain.

High-order non-reflecting boundary scheme for time-dependent waves

A new non-reflecting boundary scheme is proposed for time-dependent wave problems in unbounded domains. The linear time-dependent wave equation, with or without a dispersive term, is considered in a semi-infinite wave guide. The infinite domain is truncated via an artificial boundary B , and a high-order non-reflecting boundary condition (NRBC) is imposed on B . Then the problem is solved numerically in the finite domain bounded by B . The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. In contrast to the original formulation of the Higdon conditions, the scheme constructed here does not involve any high derivatives beyond second order. This is made possible by introducing special auxiliary variables on B . As a result, the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. The parameters appearing in the NRBC are chosen automatically via a special procedure. Numerical examples concerning a semi-infinite wave guide are used to demonstrate the performance of the new method.

Dirichlet–Neumann alternating algorithm based on the natural boundary reduction for time-dependent problems over an unbounded domain

In this paper, a new iterative algorithm to solve a time-dependent problem over an unbounded domain is suggested. This method is based on the natural boundary reduction and is suitable for solving initial boundary value problems of time-dependent wave equation over an unbounded domain. Firstly, an circular artificial boundary is introduced. Then the original unbounded domain is decomposed into a bounded domain and an exterior unbounded domain outside the artificial boundary. The natural integral equation obtained by the natural boundary reduction is used as a boundary condition on the artificial boundary. Secondly, a Dirichlet–Neumann (D–N) alternating iterative algorithm is constructed. The algorithm is equivalent to preconditioned Richardson iteration method. Thirdly, numerical studies are performed by finite element methods, and the results demonstrate the effectiveness of this algorithm. Finally, some remarks are presented.

Nonreflecting Boundary Conditions for the Time-Dependent Wave Equation

We describe a new, efficient approach to the imposition of exact nonreflecting boundary conditions for the scalar wave equation. We compare the performance of our approach with that of existing methods by coupling the boundary conditions to finite-difference schemes. Numerical experiments demonstrate a significant gain in accuracy at no additional cost.

Effect of Plasma Profile Curvature on Microwave Scattering in Reflectometry

The results of analytical solution of the time-dependent 2D full wave equation in cylindrical coordinates are presented here with application to plasma fluctuation reflectometry. We consider a family of plasma density profiles with cylindrical symmetry, which allow for a full integration of the microwave scattering problem in Born approximation. The cylindrical model is more realistic as compared to the 1D stratified plasma models previously employed in all the analytical and most numerical treatments since the plasma in the fusion toroidal devices, mirror machines and plasma processing chambers can often be considered axially symmetric on the scale relevant to microwave reflectometry. Based on the results of analytical modeling, the effects of beam refraction and scattering are studied taking into account possible asymmetry in the antenna alignment. The curvature effects on the microwave reflection and scattering are visualized and compared for the cases of plasma profiles with low and high curvatures. The reflectometer sensitivity to the small radial scale fluctuations is shown to increase in the plasma with the high curvature. Obtained expressions can be applied to the other reflectometry-related issues, such as Doppler reflectometry, phase runaway effect, and the turbulence amplitude estimation.

Nonreflecting boundary condition for the wave equation

To solve the time-dependent wave equation in an infinite two (three) dimensional domain a circular (spherical) artificial boundary is introduced to restrict the computational domain. To determine the nonreflecting boundary we solve the exterior Dirichlet problem which involves the inverse Fourier transform. The truncation of the continued fraction representation of the ratio of Hankel function, that appear in the inverse Fourier transform, provides a stable and numerically accurate approximation. Consequently, there is a sequence of boundary conditions in both two and three dimensions that are new. Furthermore, only the first derivatives in space and time appear and the coefficients are updated in a simple way from the previous time step. The accuracy of the boundary conditions is illustrated using a point source and the finite difference solution to a Dirichlet problem.