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In this article, we propose two procedures focusing on the computation of the time‐dependent convected wave equation in a free field with a uniform background flow. Both procedures are based on a framework, expended from Du et al. (SIAM J. Sci. Comput. 40 (2018), A1430–A1445.), of constructing the Dirichlet‐to‐Dirichlet (DtD)‐type discrete absorbing boundary conditions (ABCs). The first procedure is dedicated to reducing the infinite problem into a finite problem by a direct application of the framework on the finite difference discretization of the convected wave equation. However, the presence of convection terms makes the stability analysis hard to implement, which motivates us to develop the second procedure. First, the convected wave equation is transformed into a standard wave equation by using the Prandtl‐Glauert‐Lorentz transformation. After that, we obtain the DtD‐type ABC by using the above framework, and on this basis, derive an equivalent Dirichlet‐to‐Neumann‐type ABCs, which makes stability and convergence analysis easy to be obtained by the classical energy method. The effectiveness and comparison of these two procedures are investigated through numerical experiments. [ABSTRACT FROM AUTHOR]

This text discusses the behavior of solutions and the energy stability within Schwarzschild spacetimes, with a particular emphasis on the behavior of massless scalar fields under the influence of a non-rotating and spherically symmetric black hole. The stability of solutions in the proximity of the event horizon of black holes in general relativity remains an open question, especially given the difficulties introduced by minor perturbations that may resemble Kerr solutions. To address this, this work explores a simplified model, including massless scalar fields, to better understand perturbation behaviors around black holes under the Schwarzschild approach. We depart from Richard Price's work in connection with how scalar, electromagnetic, and gravitational fields behave. The tortoise coordinate transformation is considered to set the stage for numerical solutions to the wave equations. Afterward, we explore energy estimates, which are used to gauge stability and wave behavior over time. Our analysis reveals that the time evolution of the energy does not exceed twice its initial value. Further and under the assumption of initial conditions in L2-spaces, we obtain an exponential decreasing behavior in the energy time evolution. A question to continue exploring is how perturbations in L2 in the initial conditions that introduce Kerr solutions as a second-order effect in the linearized equations perturb this obtained exponential decay. [ABSTRACT FROM AUTHOR]

In this paper, novel energy-preserving finite difference schemes are presented and analyzed for one-dimensional and two-dimensional nonlinear wave equations with variable coefficients, respectively. Initially, the original differential equations are reformulated as corresponding Hamiltonian systems. Subsequently, the semi-discrete schemes are constructed using the sixth-order finite difference formula. The discrete energy method is employed to rigorously prove the energy conservation laws, convergence, and stability of the solutions for the spatial semi-discrete scheme. Next, the time derivatives in the semi-discrete schemes are discretized with the third-order strong stability preserving Runge-Kutta method, then the local truncation error of the finite difference scheme is provided. The accuracy and effectiveness of the constructed schemes, along with the theoretical analysis, are demonstrated through numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, an implicit compact finite difference (CFD) scheme was constructed to get the numerical solution for time fractional diffusion-wave equation (TFDWE), in which the time fractional derivative was denoted by Caputo-Fabrizio (C-F) sense. We proved that the full discrete scheme is unconditionally stable. We also proved that the rate of convergence in time is near to O (τ 2) and the rate of convergence in space is near to O (h 4). Test problem was considered for regular domain with uniform points to validate the efficiency and accuracy of the method. The numerical results can support the theoretical claims. [ABSTRACT FROM AUTHOR]

A theory of conversion of an intense initial wave pulse into solitons for asymptotically long evolution times has been developed using the approach based on the fact that such a transformation occurs via an intermediate stage of formation and evolution of dispersion shock waves. The number of nonlinear oscillations in such waves turns out to be equal to the number of solitons in the asymptotic state. Using the Poincaré–Cartan integral invariant theory, it is shown that the number of oscillations equal to the classical action of a particle associated with the wave packet in the vicinity of the small-amplitude edge of a dispersion shock wave remains unchanged upon a transfer by a flow described by a nondispersive limit of the nonlinear wave equations considered here. This makes it possible to formulate a generalized Bohr–Sommerfeld quantization rule that determines the set of "eigenvalues" associated with soliton physical parameters in the asymptotic state (in particular, with their velocities). In the theory, the properties of full integrability of nonlinear wave equations are not used, but the corresponding results are reproduced in this case also. The analytical results are confirmed by numerical solutions to nonlinear wave equations. [ABSTRACT FROM AUTHOR]

Seismic forward modeling is a computationally and data-intensive stage in the seismic processing workflow. By profiling the kernels of seismic forward modeling algorithms, it was observed that they need to access a wide variety of memory locations, in addition to the computational cost of performing floating-point operations for the numerical solution of wave equations. In this context, the Roofline model was used to analyze six representative computing kernels in seismic modeling on GPU environment to indicate bottlenecks in the performance and suggest improvements of these wave equation propagators. Based on this, six viscoacoustic equations were implemented using the Devito tool. Experimental data have shown that optimizations in increasing data reuse and decreasing off-chip memory traffic can significantly improve performance. [ABSTRACT FROM AUTHOR]

Lamb waves are extensively used in non-destructive tests (NDT) to detect corrosion and another defects in thin plates. Several NDT methods require to compute the phase velocity of the Lamb wave, which depends on the frequency. In this work we show the potential of the combination of the phase velocity method (PVM) with the finite element method (FEM) for computing the phase velocity dispersion curve of an ultrasonic pulse traveling in a transversally annular isotropic thin plate. The FEM-PVM is based on the numerical solution of the wave propagation equations for several selected frequencies. To solve these equations, a second order difference scheme is used to discretize the temporal variable, while spatial variables are discretized with FEM. The open software FreeFem++ is used with quadratic triangular elements to compute the displacements. The phase velocity for a given frequency is obtained from the computed displacements at few points on the top of the plate. Repeating the procedure for several frequencies we compute a sample of points that are fitted to obtain an approximation of the phase velocity dispersion curve. A deeper understanding on the behavior of the phase velocity dispersion curve with respect to the plate curvature is obtained as result of an extensive experimentation with annular isotropic plates of two materials. [ABSTRACT FROM AUTHOR]

Accurate Q (quality factor) structures can provide important constraints for characterizing subsurface hydrocarbon/water resources in exploration geophysics and interpreting tectonic evolution of the Earth in earthquake seismology. Attenuation effects on seismic amplitudes and phases can be included in forward and inverse modellings by invoking a generalized standard linear solid rheology. Compared to traditional ray-based methods, full-waveform-based adjoint tomography approach, which is based on numerical solutions of the visco-elastodynamic wave equation, has the potential to provide more accurate Q models. However, applications of adjoint Q tomography are impeded by the computational complexity of Q sensitivity kernels and by strong velocity- Q trade-offs. In this study, following the adjoint-state method, we show that the Q (P- and S-wave quality factors QP and QS) sensitivity kernels can be constructed efficiently with adjoint memory strain variables. A novel central-frequency difference misfit function is designed to reduce the trade-off artefacts for adjoint Q tomography. Compared to traditional waveform-difference misfit function, this new central-frequency approach is less sensitive to velocity variations, and thus is expected to produce fewer trade-off uncertainties. The multiparameter Hessian-vector products are calculated to quantify the resolving abilities of different misfit functions. Comparative synthetic inversion examples are provided to verify the advantages of this strategy for adjoint QP and QS tomography. We end with a 3D viscoelastic inversion example designed to simulate a distributed acoustic sensing/vertical seismic profile survey for monitoring of CO2 sequestration. [ABSTRACT FROM AUTHOR]

New numerical solution to the linear fractional-order wave equation is presented. The Liouville–Caputo sense fractional-derivative operator and Crank–Nicholson finite difference method (CN-FDM) algorithm are employed. The stability of the present technique is considered by the fractional Von Neumann stability analysis method. Special example as an application of the method is provided. The obtained results are examined to check the derived stability condition of the proposed algorithm. Computational results indicate that the present numerical algorithm is efficient and applicable for the problem under study and many other problems. [ABSTRACT FROM AUTHOR]

Even when the wave equation is a classical problem to solve in courses about hyperbolic partial differential equations, the numerical solution for this equation in irregular regions has been little studied. This is because of all the problems that could arise when solving the second-order derivative in time. This paper presents a generalized finite difference scheme applied to the numerical solution of the wave equation, solved on clouds of points for highly irregular domains. The implementation over clouds of points allows this scheme to compute satisfactory results, with minor quadratic errors, over highly irregular regions, as discussed in the present manuscript. Also, the numerical results show the accuracy obtained with a straightforward and low-cost implementation. [ABSTRACT FROM AUTHOR]

We developed a 3D, fully self-consistent model for analysis of the ultrashort THz unipolar pulse formation accompanied by its amplification in a nonequilibrium plasma channel induced in nitrogen by a femtosecond UV laser pulse. The model is based on a self-consistent numerical solution of the second-order wave equation in cylindrical geometry and the kinetic Boltzmann equation for the electron velocity distribution function (EVDF) at different points of the spatially inhomogeneous nonequilibrium plasma channel. Rapid relaxation of the electron velocity distribution function in the plasma channel results in the amplification of the leading front of the THz pulse only, while its trailing edge is not amplified or even absorbed, which gives rise to the possibility of the formation of pulses with a high degree of unipolarity. The evolution of the unipolar pulse after its transfer from the channel to open free space is analyzed in detail. [ABSTRACT FROM AUTHOR]

Inverting seismic data to build 3D geological structures is a challenging task due to the overwhelming amount of acquired seismic data, and the very-high computational load due to iterative numerical solutions of the wave equation, as required by industry-standard tools such as Full Waveform Inversion (FWI). For example, in an area with surface dimensions of 4.5 km × 4.5 km, hundreds of seismic shot-gather cubes are required for 3D model reconstruction, leading to Terabytes of recorded data. This paper presents a deep learning solution for the reconstruction of realistic 3D models in the presence of field noise recorded in seismic surveys. We implement and analyze a convolutional encoder–decoder architecture that efficiently processes the entire collection of hundreds of seismic shot-gather cubes. The proposed solution demonstrates that realistic 3D models can be reconstructed with a structural similarity index measure (SSIM) of 0.9143 (out of 1.0) in the presence of field noise at 10 dB signal-to-noise ratio. [ABSTRACT FROM AUTHOR]

In this study, we construct two explicit finite difference methods (EFDMs) for nonlinear wave equation with delay. The first EFDM is developed by modifying the standard second-order EFDM used to solve linear second-order wave equations, of which stable requirement is accepted. The second EFDM is devised for nonlinear wave equation with delay by extending the famous Du Fort-Frankel scheme initially applied to solve linear parabolic equation. The error estimations of these two EFDMs are given by applying the discrete energy methods. Besides, Richardson extrapolation methods (REMs), which are used along with them, are established to improve the convergent rates of the numerical solutions. Finally, numerical results confirm the accuracies of the algorithms and the correctness of theoretical findings. There are few studies on numerical solutions of wave equations with delay by Du Fort-Frankel-type scheme. Therefore, a main contribution of this study is that Du Fort-Frankel scheme and a corresponding new REM are constructed to solve nonlinear wave equation with delay, efficiently. [ABSTRACT FROM AUTHOR]

The paper considers the problem of inverse flood routing in reservoir operation strategy. The aim of the work is to investigate the possibility of determining the hydrograph at the upstream end based on the hydrograph required at the downstream end using simplified open channel flow models. To accomplish this, the linear kinematic wave equation, the diffusive wave equation and the linear Muskingum equation are considered. To achieve the hydrograph at the upstream end, an inverse solution of the afore mentioned equations with backward integration in the x direction is carried out. The numerical solution of the kinematic wave equation and the Muskingum equation bases on the finite difference scheme. It is shown that both these equations are able to provide satisfying results because of their exceptional properties related to numerical diffusion. In the paper, an alternative approach to solve the inverse routing using the diffusive wave model is also presented. To this end, it is described by a convolution which involves the instantaneous unit hydrograph (IUH) corresponding to the linear diffusive wave equation. Consequently, instead of a solution of partial or ordinary differential equations, the integral equation with Laguerre polynomials, used for the expansion of the upstream hydrograph, is solved. It was shown that the convolution approach is more reliable comparing to the inverse solution of the simplified models in the form of differential equations. [ABSTRACT FROM AUTHOR]

In this work, we use the phase velocity method in combination with finite element method to compute the dispersion curve for phase velocity of an ultrasonic pulse traveling in a thin isotropic plate. This method is based on the numerical solution of the wave propagation equations for several selected frequencies. To solve these equations, a second order difference scheme is used to discretize the temporal variable, while spatial variables are discretized using the finite element method. The variational formulation of the problem corresponding to a fixed value of time is obtained and the existence and uniqueness of the solution is proved. A priori error estimates in the energy norm and in the L 2 norm are also obtained. The open software FreeFem++ is used with quadratic triangular elements to compute the displacements. Numerical experiments show that the velocities computed from the approximated displacements for different frequency values are in good agreement with analytical dispersion curve. This confirms that the proposed symbiosis between finite element and phase velocity method is suitable for computing dispersion curves in more general wave propagation problems, where the geometry is complex and the material is anisotropic. [ABSTRACT FROM AUTHOR]

Here, we study the internal variable approach to viscoelasticity. First, we generalize the classical approach by introducing a fractional derivative into the equation for time evolution of the internal variables. Next, we derive restrictions on the coefficients that follow from the dissipation inequality (entropy inequality under isothermal conditions). In the example of wave propagation, we show that the restrictions that follow from entropy inequality are sufficient to guarantee the existence of the solution. We present a numerical solution to the wave equation for several values of the parameters. [ABSTRACT FROM AUTHOR]

As the key point of full-waveform inversion (FWI), numerical solution of wave equation is required to simulate the forward propagated and adjoint wavefield. When traditional regular grids are used for forward modelling, scattering artifacts may occur due to the stepped approximation of layer interfaces and rugged topography. Unstructured mesh or irregular grids method can achieve certain geometric flexibility, however, its algorithm is quite complex. Mesh-free FWI can effectively reduce the scattered artifacts under regular grids and avoid the extra computation in the process of irregular grids generation. For the implementation of mesh-free FWI method, an algorithm with fast generation of node distributions is used to discretize the velocity model, radial-basis function generated finite difference is used to realise seismic wave propagation numerical simulation, and Limited-memory BFGS (L-BFGS) algorithm is used for iteration. The mesh-free FWI method we proposed achieves flexibility of simulation region and abundant wavefield information. It reduces the storage required for FWI and illustrates the applicability of high-precision reconstruction of underground velocity in the case of rugged topography, which can provide more accurate velocity information for oil and gas exploration under complex geological conditions. [ABSTRACT FROM AUTHOR]

Numerical solution of the wave parabolic equation on a rectangular grid is analyzed when the discrete split-step Fourier transform (FT) method is used to calculate the field values in an inhomogeneous medium on the next step in the range. The goal is to find out the limiting possibilities of the FT method itself, so studies have been carried out for radio wave propagation in free space only. Two related problems have been solved. First, the minimum value of the root-mean-square error (RMSE) of the calculated field has been estimated. Second, the same has been done for the transmission coefficients of the Fourier series harmonics and of the coefficients of the artificial absorbing layer (AL), which are dependent on the parameters of the computational scheme. It is shown that the dependence of the RMSE value on the distance to the source always has a maximum. The forms of the optimal ALs differ from those used conventionally primarily by the presence of a significant imaginary component. [ABSTRACT FROM AUTHOR]

Full waveform inversion (FWI) is a procedure used to determine the elastic parameters of the Earth by reducing the misfit between observed elastodynamic wavefields and their numerically modelled counterparts. The numerical solution of the elastodynamic wave equation is computationally expensive, and its performance is typically bandwidth bound. Computing the gradient of the FWI misfit functional adds further complexity as it involves computing the zero-lag cross-correlation of two wavefields propagating in opposite temporal directions. In this paper, we utilize graphics processing units (GPUs) for their high memory bandwidth and combine two principal optimizations in order to compute FWI gradients on large models and for long simulation times. Wavefield reconstruction methods allow efficient gradient computations with minimal memory requirements and interconnection transfers. Time-space tiling techniques permit us to transcend the limited amount of GPU memory while avoiding dramatic slowdowns due to the low interconnection bandwidth. The implementation considers a task-oriented, hybrid usage of explicitly managed and Unified Memory in order to satisfy the requirements. Benchmarks demonstrate that the proposed approach is able to preserve 78–90% of the original performance, when oversubscribing the amount of physical memory available on GPUs. Comparison with existing methods highlights the benefits of the method. [ABSTRACT FROM AUTHOR]

A numerical solution of the wave equations for whistler electromagnetic waves incident on the ionosphere from above is obtained. For the calculations, we used a matrix algorithm for the approximate solution of wave equations in a stratified smoothly inhomogeneous medium and the collocation method for solving the boundary problem. Dependences of reflection coefficients R are obtained and analyzed on the frequency and angle of incidence of the wave for different seasons and time of day. At night, for waves with frequencies from 1 to 10 kHz, the values of R vary from 0.1 to 0.7. In daytime conditions, the value of R, on average, is two orders of magnitude lower and does not exceed 0.04. The smallest values of the reflection coefficient are associated with waves, the reflection of which occurs in the region of strong attenuation at heights of 80–110 km. The results obtained explain the specific features of the conditions for the excitation of the plasma magnetospheric maser. [ABSTRACT FROM AUTHOR]

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation developed in Part 1 is applied to the simulation of 1-D and 2-D test problems on regular and irregular domains. Trivial conforming and non-conforming Cartesian meshes with 3-point stencils in the 1-D case and 9-point stencils in the 2-D case are used in calculations. The numerical solutions of the 1-D wave equation as well as the 2-D wave and heat equations for a simple rectangular plate show that the accuracy of the new approach on non-conforming meshes is practically the same as that on conforming meshes for both the Dirichlet and Neumann boundary conditions. Moreover, very small distances ( 0.1 h - 10 - 9 h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not decrease the accuracy of the new technique. The application of the new approach to the 2-D problems on an irregular domain shows that the order of accuracy is close to four for the wave and heat equations and is close to five for the Poisson equation. This is in agreement with the theoretical results of Part 1 of the paper. The comparison of the numerical results obtained by the new approach and by FEM shows that at similar 9-point stencils, the accuracy of the new approach on irregular domains is two orders higher for the wave and heat equations and three orders higher for the Poisson equation than that for the linear finite elements. Moreover, the new approach yields even much more accurate results than the quadratic and cubic finite elements with much wider stencils. An example of a problem with a complex irregular domain that requires a prohibitively large computation time with the finite elements but can be easily solved with the new approach is presented. [ABSTRACT FROM AUTHOR]

Predicting the behavior of overland flow with analytical solutions to the kinematic wave equation is appealing due to its relative ease of implementation. Such simple solutions, however, have largely been constrained to applications on simple planar hillslopes. This study presents analytical solutions to the kinematic wave equation for hillslopes with modest topographic curvature that causes divergence or convergence of runoff flowpaths. The solution averages flow depths along changing hillslope contours whose lengths vary according hillslope width function, and results in a one‐dimensional approximation to the two‐dimensional flow field. The solutions are tested against both two‐dimensional numerical solutions to the kinematic wave equation (in ParFlow) and against experiments that use rainfall simulation on machined hillslopes with defined curvature properties. Excellent agreement between numerical, experimental and analytical solutions is found for hillslopes with mild to moderate curvature. The solutions show that curvature drives large changes in maximum flow rate qpeak and time of concentration tc, predictions frequently used in engineering hydrologic design and analysis. [ABSTRACT FROM AUTHOR]

This paper presents two approaches to mathematical modelling of a synthetic seismic pulse, and a comparison between them. First, a new analytical model is developed in two-dimensional Cartesian coordinates. Combined with an initial condition of sufficient symmetry, this provides a valuable check for the validity of the numerical method that follows. A particular initial condition is found which allows for a new closed-form solution. A numerical scheme is then presented which combines a spectral (Fourier) representation for displacement components and wave-speed parameters, a fourth-order Runge–Kutta integration method, and an absorbing boundary layer. The resulting large system of differential equations is solved in parallel on suitable enhanced performance desktop hardware in a new software implementation. This provides an alternative approach to forward modelling of waves within isotropic media which is efficient, and tailored to rapid and flexible developments in modelling seismic structure, for example, shallow depth environmental applications. Visual comparisons of the analytic solution and the numerical scheme are presented. [ABSTRACT FROM AUTHOR]

This paper reports the use of optical fiber Bragg-grating (FBG) sensors to monitor the stress waves generated below ground during pile driving, combined with measurements using conventional pile driving analyzer (PDA) sensors mounted at the pile head. Fourteen tubular steel piles with a diameter of 508 mm and embedded length-to-diameter ratios of 6∶20 were impact driven at an established chalk test site in Kent, United Kingdom. The pile shafts were instrumented with multiple FBG strain gauges and pile head PDA sensors, which monitored the piles' responses under each hammer blow. A high-frequency (5 kHz) fiber optic interrogator allowed a previously unseen resolution of the stress wave propagation along the pile. Estimates of the base soil resistances to driving and distributions of shaft shear resistances were found through signal matching that compared the time series of pile head PDA measurements and FBG strains measured below the ground surface. Numerical solutions of the one-dimensional wave equation were optimized by taking account of the data from multiple FBG gauges, leading to significant advantages that have potential for widespread application in cases where high-resolution strain measurements are made. [ABSTRACT FROM AUTHOR]

In this paper self-focusing of multi-Gaussian laser beam in nonlinear optical media has been investigated theoretically. Saturation of the optical nonlinearity has been incorporated through cubic-quintic model. Moment theory approach in W.K.B approximation has been invoked to find the numerical solution of nonlinear wave equation for the field of incident laser beam. Particularly the dynamical evolutions of the beam width of the laser beam with distance of propagation have been investigated in detail. It has been shown that the self-focusing of the laser beam resembles to Kepler's central force problem. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem of the damped wave equation ∂2tu − Δu + ∂tu = 0 and give sharp Lp-Lq estimates of the solution for 1 ≤ q ≤ p < ∞ (p ≠ 1) with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in (Hs ∩ Hβr) × (Hs−1 ∩ Lr) with r ∈ (1,2], s ≥ 0, and β = (n − 1)|1/2 − 1/r|, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power 1 + 2r/n, while it is known that the critical power 1 + 2/n belongs to the blow-up region when r = 1. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan by an ODE argument. [ABSTRACT FROM AUTHOR]

Abstract We consider initial value problem for semilinear damped wave equations in three space dimensions. We show the small data global existence for the problem without the spherically symmetric assumption and obtain the sharp lifespan of the solutions. This paper is devoted to a proof of the Takamura's conjecture in D'Abbicco et al. (2015) on the lifespan of solutions. [ABSTRACT FROM AUTHOR]

Abstract In this paper we investigate the following fractional order in time Cauchy problem D t α u (t) + A u (t) = f (u (t)) , 1 < α < 2 , u (0) = u 0 , u ′ (0) = u 1. The fractional in time derivative is taken in the classical Caputo sense. In the scientific literature such equations are sometimes dubbed fractional-in time wave equations or super-diffusive equations. We obtain results on existence and regularity of local and global weak solutions assuming that A is a nonnegative self-adjoint operator with compact resolvent in a Hilbert space and with a nonlinearity f ∈ C 1 (R) that satisfies suitable growth conditions. Further theorems on the existence of strong solutions are also given in this general context. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study traveling wave solutions and peakon weak solutions of the modified Camassa–Holm (mCH) equation with dispersive term 2 k u x for k ∈ R. We study traveling wave solutions through a Hamiltonian system obtained from the mCH equation by using a nonlinear transformation. The typical traveling wave solutions given by this Hamiltonian system are unbounded or multi-valued. We provide a method, called patching technic, to truncate these traveling wave solutions and patch different segments to obtain patched bounded single-valued peakon weak solutions which satisfy jump conditions at peakons. Then, we study some special peakon weak solutions constructed by the fundamental solution of the Helmholtz operator 1 − ∂ x x , which can also be obtained by the patching technic. At last, we study some length and total signed area preserving closed planar curve flows that can be described by the mCH equation when k = 1 , for which we give a Hamiltonian structure and use the patched periodic peakon weak solutions to investigate loops with peakons. [ABSTRACT FROM AUTHOR]

We introduce Devito, a new domain-specific language for implementing high-performance finite-difference partial differential equation solvers. The motivating application is exploration seismology for which methods such as full-waveform inversion and reverse-time migration are used to invert terabytes of seismic data to create images of the Earth's subsurface. Even using modern supercomputers, it can take weeks to process a single seismic survey and create a useful subsurface image. The computational cost is dominated by the numerical solution of wave equations and their corresponding adjoints. Therefore, a great deal of effort is invested in aggressively optimizing the performance of these wave-equation propagators for different computer architectures. Additionally, the actual set of partial differential equations being solved and their numerical discretization is under constant innovation as increasingly realistic representations of the physics are developed, further ratcheting up the cost of practical solvers. By embedding a domain-specific language within Python and making heavy use of SymPy, a symbolic mathematics library, we make it possible to develop finite-difference simulators quickly using a syntax that strongly resembles the mathematics. The Devito compiler reads this code and applies a wide range of analysis to generate highly optimized and parallel code. This approach can reduce the development time of a verified and optimized solver from months to days. [ABSTRACT FROM AUTHOR]

We are concerned with the initial boundary value problem of the Long-Short wave equations on the whole line. A fully discrete spectral approximation scheme is structured by means of Hermite functions in space and central difference in time. A priori estimates are established which are crucial to study the numerical stability and convergence of the fully discrete scheme. Then, unconditionally numerical stability is proved in a space of H1(R) for the envelope of the short wave and in a space of L2(R) for the amplitude of the long wave. Convergence of the fully discrete scheme is shown by the method of error estimates. Finally, numerical experiments are presented and numerical results are illustrated to agree well with the convergence order of the discrete scheme. [ABSTRACT FROM AUTHOR]

Abstract We firstly generalize a multi-term time fractional diffusion-wave equation to the multi-term variable-order time fractional diffusion-wave equation (M-V-TFD-E) by the concept of variable-order fractional derivatives. Then we implement the Chebyshev wavelets (CWs) through the operational matrix method to approximate its solution in the unit square. In fact, we apply the operational matrix of variable-order fractional derivative (OMV-FD) of the CWs to derive the unknown solution. We proceed with coupling the collocation and tau methods to reduce M-V-TFD-E to a system of algebraic equations. The important privilege of method is handling different kinds of conditions, i.e., initial-boundary conditions and Dirichlet boundary conditions, by implementing the same techniques. The convergence and error estimation of the CWs expansion in two dimensions are theoretically investigated. We also examine the applicability and computational efficiency of the new scheme through the numerical experiments. [ABSTRACT FROM AUTHOR]

More than twenty years after Murray Hodgson's "When is diffuse field theory applicable?" paper we have gathered more and more evidence that diffuse field theory is mostly a chimera. If we consider the two most important implications of the diffuse field model, i.e. uniform distribution of sound pressure level and reverberation time invariance, it is quite easy to say that such conditions are hardly ever found, based on actual measurements in a number of different spaces. Ideal sound diffusion requires ergodic and mixing conditions, which do not necessarily occur, particularly when sound absorption is unevenly distributed or rooms are not proportionate. Thus, apparently, diffuse field theory might be dismissed in favour of more accurate approaches capable of taking into account the specific nature of each space. Nowadays we have several instruments spanning from the many variations of the ray-tracing algorithm to the numerical solution of the wave equation. However, such methods rely on the measurement or estimation of other coefficients that, if not properly made, may introduce even greater inaccuracies. A critical analysis is presented here, mostly based on the author's research experience, showing that diffuse field theory still represents an important way to understand sound propagation in enclosed spaces. [ABSTRACT FROM AUTHOR]

We study a control system wtt=1ρkwxx+γw,w0t=ut,x∈0l,t∈0T, in special modified spaces of the Sobolev type. Here, ρ, k, and γ are given functions on [0, l), u 𝜖 L∞(0,T) is a control, and T > 0 is a constant. The functions ρ and k are positive on [0, l) and may tend to zero or to infinity as x → l. The growth of distributions from these spaces is determined by the growth of ρ and k as x → l. Applying the method of transformation operators, we establish necessary and sufficient conditions for the L∞-controllability and approximate L∞-controllability at a given time and for free time. [ABSTRACT FROM AUTHOR]

Wave equations with interface jump conditions have wide applications in engineering and science, for example in acoustics, elastodynamics, seismology, and electromagnetics. In this paper, an efficient non-traditional finite element method with non-body-fitted grids is proposed to solve variable coefficient wave equations with interface jump conditions. Numerical experiments show that this method is approximately second order accurate both in the Lῆ norm and L2 norm for piecewise smooth solutions. [ABSTRACT FROM AUTHOR]

The paper mainly focuses on studying nonconforming quasi-Wilson finite element fully-discrete approximation for two dimensional (2D) multi-term time fractional diffusion-wave equation (TFDWE) on regular and anisotropic meshes. Firstly, based on the Crank–Nicolson scheme in conjunction with L 1-approximation of the time Caputo derivative of order α  ∈ (1, 2), a fully-discrete scheme for 2D multi-term TFDWE is established. And then, the approximation scheme is rigorously proved to be unconditionally stable via processing fractional derivative skillfully. Moreover, the superclose result in broken H 1 -norm is deduced by utilizing special properties of quasi-Wilson element. In the meantime, the global superconvergence in broken H 1 -norm is derived by means of interpolation postprocessing technique. Finally, some numerical results illustrate the correctness of theoretical analysis on both regular and anisotropic meshes. [ABSTRACT FROM AUTHOR]

In this paper, the necessary and sufficient conditions for the existence of traveling wave solutions are derived for a class of diffusive disease-transmission models with network structures. The existence of traveling semifronts is obtained by Schauder's fixed-point theorem, and these traveling semifronts are shown to be bounded by transforming the boundedness problem into the classification problem of nonnegative solutions to a linear elliptic system on R. To overcome the reducibility problem arising in the proofs, Harnack's inequality for positive supersolutions on R is proved. [ABSTRACT FROM AUTHOR]

Abstract The current paper presents a large time step wave adding scheme (hereafter LTS-WA) originated in LeVeque's large time Godunov scheme, i.e. constructing numerical schemes via adding strong waves of discontinuity decomposition. Compared with the original scheme, we use a different strategy for wave adding and extend it to multi-dimensional cases. For rarefaction waves, we propose a grid cell decomposition method which can automatically satisfy the entropy condition and avoid nonphysical solutions. We give detailed formulae of the scheme, and make numerical experiments using scalar equations and Euler equations in one and multi-dimensional cases. Numerical results show that, besides the advantage of large time step, the new scheme has a lower numerical dissipation and a higher resolution of shocks and contact discontinuities with the increase of CFL number in certain range. Highlights • Large time step (CFL > 1). • High resolution. • Increasing resolution with CFL number. • Zero error at integer CFL number in linear case. [ABSTRACT FROM AUTHOR]

In this paper, the higher-order nonlinear Schrödinger equation, which can be widely used to describe the dynamics of the ultrashort pulses in optical fibers, is under investigation. By means of the modified Darboux transformation, the hierarchies of breather wave and rogue wave solutions are generated from the trivial solution. Furthermore, the main characteristics of the breather and rogue waves are graphically discussed. The results show that the extreme behavior of the breather wave yields the rogue wave for the higher-order nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

We consider a class of semilinear wave equations with memory and Neumann boundary conditions, being subject to frictional dissipation. As can be seen, their solutions have different properties from those in the case of Dirichlet boundary conditions (or Dirichlet-Neumann boundary conditions). We prove for some nonlinear terms with growth exponent that all solutions decay uniformly and at least at the polynomial rate , when the memory kernel decays exponentially or polynomially (with large enough degree r); some other decay rates, depending on both q and r, are also derived, when r is not large enough. Moreover, we show a large class of solutions decaying exactly at the rate , in the case of the memory kernels decaying exponentially. [ABSTRACT FROM AUTHOR]

We give an analytical justification of the determination of an important parameter involved in Feng and Wu's solution of the Helmholtz equation by an Interior Penalty Discontinuous Galerkin (IPDG) method (Feng and Wu, 2009). This parameter was determined in this reference from a large number of numerical tests for a mesh composed of equal equilateral triangles at a fixed frequency. It plays an essential role in the reduction the dispersion error for a polynomial approximation of degree 1. We show how this justification makes it possible to slightly improve it and to handle other structured meshes. The analytical results are evaluated by several numerical experiments, showing in particular that the parameter determined in principle for equal equilateral triangles also eliminates dispersion for unstructured, well-smoothed meshes. • Numerical solution of the wave equation in the frequency domain. • Use of Interior Penalty Discontinuous Galerkin methods. • Use lower degree approximation. • Build methods that considerably reduce the dispersion. [ABSTRACT FROM AUTHOR]

In this paper, we present an adaptive spectral projection based finite element method to numerically approximate the solution of the wave equation with memory. The adaptivity is not restricted to the mesh (h p -adaptivity), but it is also applied to the size of the computed spectrum (k -adaptivity). The meshes are refined using a residual based error estimator, while the size of the computed spectrum is adapted using the L 2 norm of the error of the projected data. We show that the approach can be very efficient and accurate. [ABSTRACT FROM AUTHOR]

In this paper, we present a nonlocal perfectly matched layer (PML) for the nonlocal wave equation in two dimensions, and design numerical discretization to solve the reduced PML problem on a bounded domain. The core idea of designing the nonlocal PML is to transform the wave equation into a nonlocal Helmholtz equation via the Laplace transform, and then analytically continue it into the complex plane. Unlike the original nonlocal operator, the new nonlocal operator in the nonlocal PML equation is complex-valued and defined by time convolution. We numerically approximate the nonlocal PML operator by combining a quadrature-based finite difference scheme and Talbot's contour, and finally present the full discretization. The accuracy and effectiveness of our approaches are illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

We show that an energy-critical wave map into the sphere that has energy just above the degree one harmonic maps and that blows up in finite time asymptotically decouples into a regular part plus a traveling wave with small momentum, in the energy space. In particular, the only possible form of energy concentration is through the concentration of traveling waves. This is often called "quantization of energy" at blow up. The main new tool is a channel of energy type inequality for outgoing small energy wave maps similar to the one proved in our previous work [ 14 ] on outgoing solutions of the energy-critical wave equation. We also give a brief review of important background results in the subcritical and critical regularity theory for the two dimensional wave maps from [ 35 – 37, 47, 48, 51, 56, 57 ]. [ABSTRACT FROM AUTHOR]

Abstract This paper considers pseudo-impulsive numerical solutions to the forward-speed diffraction problem, as derived from classical linearized potential flow theory. Both head- and following-seas cases are treated. Fourth-order finite-difference approximations are applied on overlapping, boundary-fitted grids to obtain solutions using both the Neumann-Kelvin and the double-body flow linearizations of the problem. A method for computing the pseudo-impulsive incident wave forcing in finite water depth using the Fast Fourier Transform (FFT) is presented. The pseudo-impulsive scattering solution is then Fourier transformed into the frequency domain to obtain the wave excitation forces and the body motion response. The calculations are validated against reference solutions for a submerged circular cylinder and a submerged sphere. Calculations are also made for a modern bulk carrier, showing good agreement with experimental measurements. [ABSTRACT FROM AUTHOR]