Solving Wave Equations Research Articles

Solving wave equations in the space of Schwartz distributions: the beauty of generalised functions in physics

Solving wave equations in the space of Schwartz distributions: the beauty of generalised functions in physics

Frozen Gaussian sampling for scalar wave equations

In this article, we introduce the frozen Gaussian sampling (FGS) algorithm to solve the scalar wave equation in the high-frequency regime. The FGS algorithm is a Monte Carlo sampling strategy based on the frozen Gaussian approximation, which greatly reduces the computation workload in wave propagation and reconstruction. In this work, we propose feasible and detailed procedures to implement the FGS algorithm to approximate scalar wave equations with Gaussian and WKB initial conditions respectively. For both initial data cases, we rigorously analyze the error of applying this algorithm to wave equations of dimensionality d ≥ 3. In Gaussian initial data cases, we prove that the sampling error due to the Monte Carlo method is independent of the typical wave number. We also derive a quantitative bound of the sampling error in WKB initial data cases. Finally, we validate the performance of the FGS and the theoretical estimates about the sampling error through various numerical examples, which include using the FGS to solve wave equations with both Gaussian and WKB initial data of dimensionality d = 1, 2, and 3.

Enabling uncertainty quantification in a standard full-waveform inversion method using normalizing flows

To maximize the utility of seismic imaging and inversion results, we need to compute not only a final image but also quantify the uncertainty in the image. Although the most thorough approach to quantify the uncertainty is to use a method such as Markov chain Monte Carlo, which systematically samples the entire posterior distribution, this is often inefficient, and not all applications require a full representation of the posterior. We use normalizing flows (NFs), a machine learning technique to perform uncertainty quantification (UQ) in full-waveform inversion (FWI), specifically for time-lapse data. As with any machine learning algorithm, the NF learns only the mapping from the part of the prior spanned by the training data to the distribution of final models spanned by the training data. Here, we make use of this property to perform UQ efficiently by learning a mapping from the prior to the distribution that characterizes the model perturbations within a specific range. Our approach involves using a range of starting models paired with final models from a standard FWI as training data. Although this does not capture the full posterior of the FWI problem, it enables us to quantify the uncertainties associated with updating from an initial to a final model. Because our target is to perform UQ for time-lapse imaging, we use a local wave-equation solver that allows us to solve the wave equation in a small subset of our entire model, thereby keeping computational costs low. Numerical examples demonstrate that incorporating the training step for NF provides a distribution of model perturbations, which is dependent on a designated prior, to quantify the uncertainty of FWI results.

Full wave modeling of radio-frequency beams in tokamaks in the electron cyclotron frequency range

Simulation of full wave, without paraxial approximation, high-resolution solution of wave equations in frequency domain in the electron cyclotron resonance (ECR) frequency range for realistic Tokamak plasma parameters became possible by using recently formulated hybrid iterative algorithm [Svidzinski et al., Phys. Plasmas 25, 082509 (2018)] for numerically solving discretized wave equations. This approach combines time evolution and iterative relaxation techniques into iteration cycles. This algorithm is implemented in 2D code FullWave, solving wave equations in Tokamaks in cold and hot plasma models, and it has been tested in 3D full wave iterative RF beams simulation tool, which is presently being developed to model 3D ECRH RF beams in fusion devices using dynamic grid adaptation. The results of 2D full wave modeling, assuming specified toroidal mode number, of ECRH RF beams in DIII-D plasma, performed in the cold and hot plasma models for outboard and top launch scenarios using FullWave are presented. Nonlocal hot plasma response model, based on accurate numerical solution of linearized Vlasov equation, is used to model beam propagation and absorption in the 2nd electron cyclotron harmonic region. Demonstration of capability of the hybrid iterative algorithm to model ECRH RF beams in 3D is made by simulating a substantial part of realistic beam in DIII-D, launched from outboard side of the machine. All relevant physics of RF beam propagation, most of which is not captured in paraxial approximation, such as beam's divergence, interference between the X and O modes in the beam, X-O mode conversion, beam splitting into the X and O mode beams, transformation of beam's cross section, and absorption at the 2nd electron cyclotron harmonic, is captured in the simulations. A numerical technique to find an optimal beam polarization at the launcher to launch a nearly pure X or O mode beam in plasma is developed and tested.

A mixed Fourier-variational approach to solve differential or integro-differential wave equations for magnetised plasmas

The All ORders Spectral Algorithm (AORSA) wave equation solver by Jaeger (Jaeger et al 2001 Phys. Plasmas 8 1573) solves the integro-differential wave equation relevant for the radio frequency (RF) domain and for fusion-relevant conditions in tokamaks or stellarators, retaining all finite Larmor radius corrections by substituting the continuous Fourier integrals by a sum over a discrete set of modes. Its strength is also its weakness: the simplicity of the method results in significant computational effort, a full matrix needing to be inverted to solve the associated linear system. Based on the notion that modes are gradually more independent if their eigenvalues differ, the present paper proposes a straightforward numerical method to partly alleviate this need, allowing to substitute the full system matrix by a banded one. The adopted method can be applied to a wide variety of equations. A few 1D examples—of relevance for solving the wave equation in the RF domain of frequencies—are provided: the tunneling equation is used to illustrate the potential of the method, and the all-FLR wave equation (retaining all Finite Larmor Radius corrections in the dielectric response) adopted by Jaeger is solved comparing the solutions found to those based on simpler models (a cold plasma and a ‘tepid plasma’ - i.e. a kinetic model truncated at zero order in Larmor radius—description).

Wave amplitude gain within wedge waveguides through scattering by simple obstacles.

Wave confinement, e.g., in waveguides, gives rise to a huge number of distinct phenomena. Among them, amplitude gain is a recurrent and relevant effect in undulatory processes. Using a general purpose protocol to solve wave equations, the boundary wall method, we demonstrate that for relatively simple geometries, namely, a few leaky or opaque obstacles inside a θ wedge waveguide (described by the Helmholtz equation), one can obtain a considerable wave amplification in certain spatially localized regions of the system. The approach relies on an expression for the wedge waveguide exact Green's function in the case of θ=π/M (M=1,2,...), derived through the method of images allied to group theory concepts. The formula is particularly amenable to numerical calculations, greatly facilitating simulations. As an interesting by-product of the present framework, we are able to obtain the eigenstates of certain closed shapes (billiards) placed within the waveguide, as demonstrated for triangular structures. Finally, we briefly discuss possible concrete realizations for our setups in the context of matter and electromagnetic (for some particular modes and conditions) waves.

Immersed Finite Element Method for Solving Wave Equations with Discontinuous Coefficients

Immersed Finite Element Method for Solving Wave Equations with Discontinuous Coefficients

Asymptotic expansion of the error of the numerical method for solving wave equation with functional delay

A wave equation with functional delay is considered. The problem is discretized. Constructions of the difference method with weights with piecewise linear interpolation are given. A basic method with weights with piecewise cubic interpolation is constructed. The order of the residual is studied without interpolation of the basic method, and the expansion coefficients of the residual with respect to time-steps and space-steps are written out. It is proved that the weighted method with piecewise cubic interpolation converges with order 2 in the energy norm. An equation is written for the main term of the asymptotic expansion of the global error of the basic method. Under certain assumptions, the validity of the application of the Richardson extrapolation procedure is substantiated, and the corresponding numerical method is constructed, that has the fourth order of convergence with respect to time-steps and space-steps. The validity of Runge's formulas for practical estimation of the error is proved. The results of numerical experiments on a test example are presented.

Scalar Perturbations of Gravitational Collapse under Homotopy Perturbation Method: A Critical Study on the Rectangular and Price Potential

AbstractIn order to unveil different features of gravitational waves due to massive objects like black hole etc., researchers primarily attempt to find the corresponding quasi normal modes from the relevant wave equations. In the present article, homotopy perturbation method (HPM) is employed to obtain the wave functions by solving wave equations for one‐dimensional potential barriers. Under this scheme (i) for constant potential barrier a generalized solution which reduces to the known solution as done by Chandrasekhar and Detweiler [Proc. R. Soc. Lond. A 344 (1975) 441] is produced, (ii) the wave equation for the Price potential [Phys. Rev. D 5 (1972) 2419; Phys. Rev. D 5 (1972) 2439] by HPM is solved. The wave functions (graphically) as well as quasi‐normal modes to those of the Press solution (1973) are compared. The present investigation is important in connection to further study related to the Zerilli potential and Regge‐Wheeler potential in case of black hole.

From ground motion simulations to landslide occurrence prediction

Ground motion simulations solve wave equations in space and time, thus producing detailed estimates of the shaking time series. This is essentially uncharted territory for geomorphologists, for we have yet to understand which ground motion (synthetic or not) parameter, or combination of parameters, is more suitable to explain the coseismic landslide distribution. To address this gap, we developed a method to select the best ground motion simulation using a combination of Synthetic Aperture Radar Interferometry (InSAR) and strong motion data. Upon selecting the best simulation, we further developed a method to extract a suite of intensity parameters, which we used to analyse coseismic landslide occurrences taking the Gorkha earthquake (M7.8, 25th April 2015) as a reference. Our results show that beyond the virtually unanimous use of peak ground acceleration, velocity, or displacement in the literature, different shaking parameters could play a more relevant role in landslide occurrences. These parameters are not necessarily the product of peak ground motion but are linked to the total displacement, frequency content, and shaking duration; elements too often neglected in geomorphological analyses. This, in turn, implies that we have yet to fully acknowledge the complexity of the interactions between full waveforms and hillslope responses.

Propagating analysis of Airy beams via complex phase space and ray method.

A complex phase space and ray method is proposed to present intuitive interpretation of unique intensity profiles, parabolic accelerating trajectories, and distinctive phase shifts of Airy beam and its exponentially decaying version (i.e., finite-energy Airy beam). In the complex phase space, finite-energy Airy beam manifests itself as a complex parabolic phase space curve (PSC) which represents a cluster of light rays with complex wave vectors. The complex ray cluster converges to a complex parabolic caustic curve in complex coordinate space. For infinite-energy Airy beam, phase space, PSC, light ray and caustic curve change to real values. In the paraxial condition, Airy beams can maintain parabolic form of PSC and keep constant ray density, which guarantees the non-diffraction property of Airy beam and approximate non-diffraction property of finite-energy Airy beam. From the evolution of vertexes of parabolic PSC along a parabolic trajectory in phase space, one can also give the parabolic caustic curve for Airy beam and the complex parabolic caustic curve for exponentially decaying Airy beam. Further, the special phase and decay factors in the propagating solution of complex amplitude of Airy beams can be directly derived from the phase shifts of light ray cluster along transverse and longitudinal directions. The proposed phase space and ray method can present intuitive comprehension to distinctive propagating characteristics of Airy beams including intensity and phase, without resort to solving wave equation or diffracting integral formula.

Using the Stix finite element RF code to investigate operation optimization of the ICRF antenna on Alcator C-Mod

As the Ion Cyclotron Radio Frequency range (ICRF) heating becomes more favorable in fusion devices, the urgency of predicting and mitigating impurity generation that arises from it becomes more pressing. In the ICRF regime, rectified Radio Frequency (RF) sheaths are known to form at antenna and material edges that influence negative effects like sputtering and a decrease in heating efficiency. Methods to mitigate the formation of these RF sheaths through RF image currents cancellation have been experimentally studied. A power-phasing scan done on Alcator C-Mod in which the amount of power on the two inner straps (P in) versus the total 4 straps (P tot) was varied showed a minimization of enhanced potentials between Pin/Ptot∼ 0.7–0.9 while impurities were minimized for Pin/Ptot∼ 0.5–0.8. New capabilities in the realm of representing the RF sheath numerically now allow for these experiments to be simulated. Given the size of the sheath relative to the scale of the device, it can be approximated as a Boundary Condition (BC). A new parallelized cold-plasma wave equation solver called Stix implements a non-linear sheath impedance model BC formulated by Myra et al (2015 Phys. Plasmas 22 062507) through the method of finite elements using the MFEM library [http://mfem.org]. It is seen that Stix shows qualitative agreement with the measured C-Mod enhanced potentials.

A modified Bohr’s quantization rule to estimate the energy levels of relativistic systems

Abstract Estimation of quantum energy levels is crucial to understand the physical systems in both non relativistic and relativistic domains. Here we put forward a new method to deduce quantized energy levels of some simple relativistic quantum systems without solving wave equations but using Bohr’s quantization rule along with a few suitable modifications. The results obtained are in good agreement with formal calculations available in the literature.

Multi-material described metasurface solar absorber design with absorption prediction using machine learning models

As an integral part of a STPV system, a solar absorber is essential to absorb most of the energy of the incident solar spectral irradiance. Metasurfaces have revolutionized contemporary research in optics and photonics and can benefit from the unique thermostable properties of refractory materials as well. We propose a metasurface-based solar absorber having a simple symmetric square-ring structure employing seven different refractory materials with an intent to boost the efficiency of the solar energy system. The proposed design exhibits broadband absorption for the visible regime and bears similarity to AM 1.5 solar spectral irradiance indicating the effectiveness of the design. The absorption is optimized in every case by varying the defining geometrical descriptors. The presented designs have in addition shown a high absorption rate over a range of incidence angles. Traditionally, the desired optical responses used to be obtained through electromagnetic simulation software by solving wave equation iteratively. This process spans hundreds of numerical calculations making it a time-consuming and computation-intensive strategy. Instead of following the classical numerical approaches for obtaining the EM responses, the data-driven intelligent approach based on machine learning is applied to predict the optical response. The algorithms including decision tree, K-nearest neighbor, linear and polynomial regression have been investigated for the due purpose of replacing the conventional simulations, with decision tree algorithm performing exceptionally fast and accurate. The trained models have shown an excellent predictability for broadband absorption (300–1200 nm), with MSE of the model equal to only 6.38 × 10−5 and R2 score of 0.9999. Further, we have trained the inverse model by employing a dimensionality reduction technique “PCA” on optical response vectors to predict the geometrical parameters of meta-atom yielding a min of 0.0 MSE and R2 score equal to 0.9999. The modeling procedure and numerical calculations are performed in a matter of seconds over the whole dataset especially when PCA with 10 components is applied. The presented approach bids a novel and effective methodology to speed up the on-demand design of intricate metasurfaces and optical structures for green energy applications.

Improved least-squares migration through double-sweeping solver

Based on a recently developed approximate wave-equation solver, we have developed a methodology to reduce the computational cost of seismic migration in the frequency domain. This approach divides the domain of interest into smaller subdomains, and the wavefield is computed using a sequential process to determine the downward- and upward-propagating wavefields — hence called a double-sweeping solver. A sequential process becomes possible using a special approximation of the interface conditions between subdomains. This method is incorporated into the least-squares migration framework as an approximate solver. The associated computational effort is comparable to one-way wave-equation approaches, yet, as illustrated by the numerical examples, the accuracy and convergence behavior are comparable to that of the full-wave equation.

Multiscale methods for solving wave equations on spatial networks

We present and analyze a multiscale method for wave propagation problems, posed on spatial networks. By introducing a coarse scale, using a finite element space interpolated onto the network, we construct a discrete multiscale space using the localized orthogonal decomposition (LOD) methodology. The spatial discretization is then combined with an energy conserving temporal scheme to form the proposed method. Under the assumption of well-prepared initial data, we derive an a priori error bound of optimal order with respect to the space and time discretization. In the analysis, we combine the theory derived for stationary elliptic problems on spatial networks with classical finite element results for hyperbolic problems. Finally, we present numerical experiments that confirm our theoretical findings.

Parametrization-free locally-conformal perfectly matched layer method for finite element solution of Helmholtz equation

We present a novel Locally-Conformal Perfectly Matched Layer (LCPML) method for mesh truncation in the Finite Element Method (FEM), to solve wave equations derived from Maxwell's equations. The original LCPML method was proposed by Ozgun and Kuzuoglu in 2006 [16] by utilizing the concept of complex elements in which nodal coordinates of elements are replaced by complex coordinates obtained by using a complex coordinate-transformation. In the proposed LCPML method, named as LCPML-log method, a more efficient coordinate-transformation is used together with new FEM formulation and implementation. The proposed method offers the advantage that only a few (or even a single) PML layer is sufficient to get reliable results. In addition, it does not require any PML parameter tuning, and hence, the decaying wave behavior in PML is achieved self-adaptively. The numerical results of some detailed and systematic performance analyses are also presented.

The Use of Fourier Transform in Heat Equation and Wave Equation

In the solve for the equation for heat transfer and wave movement, using Fourier Transform to simplify the differential equation and solve it. With the periodic property of the wave including standing wave and vibrating string and heat equation and its stable solution, it can provide many new properties of the equation. Then using the same method to solve the heat equation in space, it gives a similar equation like the plane heat transfer equation. The passage mainly discussed about using Fourier transform to solve standing waves equation heat equation in steady state and in the time dependent state. The general way of solving wave equation both standing wave and traveling wave) is finding differential equation for a half period, expending it into the R, using Fourier transform to solve the equation.

On the dynamics of the nonautonomous multi-soliton, multi-lump waves and their collision phenomena to a (3+1)-dimensional nonlinear model

One of the most popular ways to solve wave equations and explain nonlinear physical phenomena is with nonautonomous soliton solutions. This work aims to analyze the (3+1)-dimensional variable-coefficient Kudryashov–Sinelshchikov model, which explains pressure waves in liquid with flocs. Fluid dynamics, a subfield of fluid mechanics used in physics and engineering, is the study of how liquids and gases flow. There are many subfields within it, including aerodynamics and hydrodynamics. There are several uses for fluid dynamics, namely quantifying forces and moments. Using Hirota direct method, we auspiciously provide multiple solitons and M-lump solutions to this equation. We use specified input variables to accentuate the physical traits of the results obtained via two-, three-dimensional, and contour graphics since doing so is significant. The findings are applied to exemplify the physical traits of lump solutions and the collision-related elements of various nonlinear physical processes.

Seamless GPU Acceleration for C++-Based Physics with the Metal Shading Language on Apple’s M Series Unified Chips

Abstract The M series of chips produced by Apple has proven a capable and power-efficient alternative to mainstream Intel and AMD ×86 processors for everyday tasks. In addition, the unified design integrating the central processing and graphics processing unit (GPU), have allowed these M series chips to excel at many tasks with heavy graphical requirements without the need for a discrete GPU) in some cases even outperforming discrete GPUs. In this work, we show how the M series chips can be leveraged using the Metal Shading Language (MSL) to accelerate typical array operations in C++. More important, we show how the usage of MSL avoids the typical complexity of compute unified device architecture (CUDA) or OpenACC memory management by allowing the central processing unit (CPU) and GPU to work in unified memory. We demonstrate how performant the M series chips are on standard 1D and 2D array operations such as array addition, single-precision A·X plus Y, and finite-difference stencils, with respect to serial and OpenMP-accelerated CPU code. The reduced complexity of implementing MSL also allows us to accelerate an existing elastic wave equation solver (originally based on OpenMP-accelerated C++) while retaining all CPU and OpenMP functionality without modification. The resulting performance gain of simulating the wave equation is near an order of magnitude for large domain sizes. This gain attained from using MSL is similar to other GPU-accelerated wave-propagation codes with respect to their CPU variants but does not come at much increased programming complexity that prohibits the typical scientific programmer to leverage these accelerators. This result shows how unified processing units can be a valuable tool to seismologists and computational scientists in general, lowering the bar to writing performant codes that leverage modern GPUs.