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The tensor-train mimetic finite difference method for three-dimensional Maxwell’s wave propagation equations
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A generalized Mimetic Finite Difference method and Two-Point Flux schemes over Voronoi diagrams
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A dispersion optimized mimetic finite difference method for Maxwell's equations in metamaterials.
Abstract not provided.
A Dispersion Minimized Mimetic Finite Difference Method for Maxwell's Equations in Linear Dispersive Dielectrics.
Abstract not provided.
Parallel diffusion operator for magnetized plasmas with improved spectral fidelity
Diffusive transport processes in magnetized plasmas are highly anisotropic, with fast parallel transport along the magnetic field lines sometimes faster than perpendicular transport by orders of magnitude. This constitutes a major challenge for describing non-grid-aligned magnetic structures in Eulerian (grid-based) simulations. Here, the present paper describes and validates a new method for parallel diffusion in magnetized plasmas based on the anti-symmetry representation [Halpern and Waltz, Phys. Plasmas 25, 060703 (2018)]. In the anti-symmetry formalism, diffusion manifests as a flow operator involving the logarithmic derivative of the transported quantity. Qualitative plane wave analysis shows that the new operator naturally yields better discrete spectral resolution compared to its conventional counterpart. Numerical simulations comparing the new method against existing finite difference methods are carried out, showing significant improvement. In particular, we find that combining anti-symmetry with finite differences in diagonally staggered grids essentially eliminates the so-called “artificial numerical diffusion” that affects conventional finite difference and finite volume methods.
Enhancing photoionization rate calculations in low-temperature plasmas using spectral methods
Photoionization plays a central role in the development of streamer discharges and other non-equilibrium plasma phenomena. It creates seed electrons, which are essential for positive streamer propagation, allowing the ionization front to move forward. Because of this, accurate modeling of photoionization is very important for predicting streamer behavior and plasma evolution. The photoionization process in air (N 2 – O 2 mixture) is often described by the Zheleznyak model (1982). This model is usually solved through Helmholtz-type equations that approximate the Zheleznyak photoionization model (Zheleznyak et al. 1982) as Partial Differential Equations (PDEs). Conventional numerical methods, such as the Finite Difference Method (FDM) or Finite Volume Method (FVM), are widely used to solve these equations. Although they are prevalent, the computational cost of these methods due to their need for matrix operations and iterative solver is demanding. To address this challenge, this work develops a spectral solver based on the Fast Fourier Transform (FFT) combined with Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) to calculate the photoionization rate efficiently in an axisymmetric cylindrical domain. This method naturally satisfies the boundary conditions used in the model and converts the PDE into algebraic ones in spectral space. Thus, avoids the need for iterative matrix solvers. When compared with FDM results, it is demonstrated that the new solver not only maintains accuracy, but also reduces the computational cost, showing a performance increase of approximately 100 compared to FDM over a wide range of problem sizes. The method is parallelized using Message Passing Interface (MPI) and has been integrated into a fluid plasma model for streamer simulation. Here, this FFT-based approach provides a fast and reliable alternative for calculating photoionization in fluid models, helping large-scale plasma simulations run faster and efficiently, and allows higher-resolution simulation without extra computational cost.
Combination of the discontinuous Galerkin method with finite differences for simulation of seismic wave propagation
We present an algorithm for the numerical simulation of seismic wave propagation in models with a complex near surface part and free surface topography. The approach is based on the combination of finite differences with the discontinuous Galerkin method. The discontinuous Galerkin method can be used on polyhedral meshes; thus, it is easy to handle the complex surfaces in the models. However, this approach is computationally intense in comparison with finite differences. Finite differences are computationally efficient, but in general, they require rectangular grids, leading to the stair-step approximation of the interfaces, which causes strong diffraction of the wavefield. In this research we present a hybrid algorithm where the discontinuous Galerkin method is used in a relatively small upper part of the model and finite differences are applied to the main part of the model.
Theoretical analysis and numerical simulation for a hyperbolic equation with Dirichlet and acoustic boundary conditions
This paper is concerned with a theoretical and numerical study for the initial-boundary value problem for a linear hyperbolic equation with variable coefficient and acoustic boundary conditions. On the theoretical results, we prove the existence and uniqueness of global solutions, and the uniform stability of the total energy. Numerical simulations using the finite element method associated with the finite difference method are employed, for one-dimensional and two-dimensional cases, to validate the theoretical results. In addition, numerically the uniform decay rate for energy and the order of convergence of the approximate solution are also shown.
An Early Investigation of the HHL Quantum Linear Solver for Scientific Applications
In this paper, we explore using the Harrow–Hassidim–Lloyd (HHL) algorithm to address scientific and engineering problems through quantum computing, utilizing the NWQSim simulation package on a high-performance computing platform. Focusing on domains such as power-grid management and climate projection, we demonstrate the correlations of the accuracy of quantum phase estimation, along with various properties of coefficient matrices, on the final solution and quantum resource cost in iterative and non-iterative numerical methods such as the Newton–Raphson method and finite difference method, as well as their impacts on quantum error correction costs using the Microsoft Azure Quantum resource estimator. We summarize the exponential resource cost from quantum phase estimation before and after quantum error correction and illustrate a potential way to reduce the demands on physical qubits. This work lays down a preliminary step for future investigations, urging a closer examination of quantum algorithms’ scalability and efficiency in domain applications.
Compact finite difference relaxation method for chaotic and hyperchaotic initial value systems
In this paper, we present a new application of higher order compact finite differences to solve nonlinear initial value problems exhibiting chaotic behaviour. The method involves dividing the domain of the problem into multiple sub-domains, with each sub-domain integrated using higher order compact finite difference schemes. The nonlinearity is dealt using a Gauss–Seidel-like relaxation. The method is, therefore, referred to as the multi-domain compact finite difference relaxation method (MD-CFDRM). In this new application, the MD-CFDRM is used to solve famous chaotic systems and hyperchaotic systems. The main advantage of the new approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method. The results are compared with spectral-based multi-domain method.
Highly tunable compact polarization rotator based on silicon on insulator platform
A novel design of an ultra-compact polarization rotator (PR) based on silicon on insulator platform infiltrated with liquid crystal is reported and studied. The anisotropic property of the liquid crystal material leads to a high modal hybridness, which improves the power exchange between the two orthogonal polarized modes. Therefore, tunable and high polarization conversion can be achieved over a short propagation distance. The proposed PR is analyzed by using full-vectorial finite-difference method and full-vectorial finite-difference beam propagation method. At an operating wavelength of 1.55 μm, the PR has a compact device length of 5.4 μm with 14 dB extinction ratio (ER) and 0.15 dB insertion loss. Moreover, the suggested PR offers a relatively broad bandwidth of (1340–1685) nm with an overall ER better than 10 dB. Further, the polarization conversion ratio can be tuned by controlling the rotation angle of the nematic liquid crystal molecules via an external electric field. Furthermore, the fabrication tolerance study shows that the reported PR has high robustness to the fabrication errors.
Sensitivity Analysis of Hydraulic Head to Locations of Model Boundaries
Sensitivity analysis is an important component of many model activities in hydrology. Numerous studies have been conducted in calculating various sensitivities. Most of these sensitivity analysis focus on the sensitivity of state variables (e.g. hydraulic head) to parameters representing medium properties such as hydraulic conductivity or prescribed values such as constant head or flux at boundaries, while few studies address the sensitivity of the state variables to some shape parameters or design parameters that control the model domain. Instead, these shape parameters are typically assumed to be known in the model. In this study, based on the flow equation, we derive the equation (and its associated initial and boundary conditions) for sensitivity of hydraulic head to shape parameters using continuous sensitivity equation (CSE) approach. These sensitivity equations can be solved numerically in general or analytically in some simplified cases. Finally, the approach has been demonstrated through two examples and the results are compared favorably to those from analytical solutions or numerical finite difference methods with perturbed model domains, while numerical shortcomings of the finite difference method are avoided.
A finite difference informed random walker (FDiRW) solver for strongly inhomogeneous diffusion problems
In nature, many complex multi-physics coupling problems exhibit strong diffusivity inhomogeneity. For instance, in the context of radionuclide absorption by porous wasteform materials within a flowing waste stream, the difference of species’ diffusivity in solid and liquid phases spans by 3~8 orders of magnitude. To solve the diffusion equations with strongly inhomogeneous diffusivity, traditional discretization-based methods, such as the Finite Difference Method (FDM), require infinitesimally small time steps (<10 -10 ) as high spatial resolutions are employed in most microstructure evolution processes, leading to prohibitively high computational costs. Here, this work developed an integrated numerical approach (FDiRW: Finite Difference informed Random Walk) to tackle this challenge. The idea is that utilizing the Random Walk concept, the fast diffusion is modeled as a superposition of point source’s solution for a concentration distribution while FDM is used to obtain the point source’s solution at each node. A mesh-coarsening algorithm is developed to generate an exclusive coarse mesh for FDiRW approach to maximize its efficiency. The effectiveness of the coarse mesh-based FDiRW approach is validated by benchmarking Finite Difference solutions. Numerical results demonstrated that FDiRW achieves a remarkable 1000x computational efficiency improvement over FDM while preserving desired accuracy for a medium-sized model of 192 × 192 × 192 grids. Finally, as models scale up, a floating-point operations (PLOPs) analysis of the FDiRW algorithm reveals that its computational complexity grows quadratically in terms of the number of nodes employed in computation.
Acoustic Codes in 2D Spherical Coordinate
Finite-difference methods are widely used to simulate infrasound propagation in the atmosphere. Flexibility of finite-difference scheme allows implementation of highly heterogeneous media for sound propagation as well as complex source models for sound generation. While full 3-D finite-difference methods have been utilized for local infrasound propagation with pronounced topography, 2-D modeling approach has been preferred for regional and global propagation as full 3-D methods generally require enormous computational resources. Infrasound propagation is often simulated with a second-order finite difference scheme. This lowest-order finite-difference scheme is robust and straightforward to implement complex boundary conditions, but the solution includes large error with numerical dispersion and dissipation. This large numerical error may make the second-order finite-difference not optimal for long range infrasound propagation modeling as the numerical dispersion degrades the accuracy of the solution unacceptably. Here, we developed a high-order finite-difference solver for long-range infrasound simulation. The high-order scheme is particularly popular for linear wave modeling in aeroacoustics owing to its low-dispersive and low-dissipative behavior. We develop and evaluate a high-order finite difference scheme in 2-D axisymmetric coordinates. The axisymmetry allows to approximate 3-D spherical sound propagation and amplitude attenuation by a 2-D method. AC2Dr is developed to simulate infrasound propagation in realistic atmosphere, but can be used for linear acoustic waves in general materials with background flow. AC2Dr in an axisymmetric coordinates allows for spherical radiation of acoustic waves from compact sources.
Finite-difference time-domain methods
The finite-difference time-domain (FDTD) method is a widespread numerical tool for full-wave analysis of electromagnetic fields in complex media and for detailed geometries. Applications of the FDTD method cover a range of time and spatial scales, extending from subatomic to galactic lengths and from classical to quantum physics. Technology areas that benefit from the FDTD method include biomedicine — bioimaging, biophotonics, bioelectronics and biosensors; geophysics — remote sensing, communications, space weather hazards and geolocation; metamaterials — sub-wavelength focusing lenses, electromagnetic cloaks and continuously scanning leaky-wave antennas; optics — diffractive optical elements, photonic bandgap structures, photonic crystal waveguides and ring-resonator devices; plasmonics — plasmonic waveguides and antennas; and quantum applications — quantum devices and quantum radar. This Primer summarizes the main features of the FDTD method, along with key extensions that enable accurate solutions to be obtained for different research questions. Additionally, hardware considerations are discussed, plus examples of how to extract magnitude and phase data, Brillouin diagrams and scattering parameters from the output of an FDTD model. Furthermore, the Primer ends with a discussion of ongoing challenges and opportunities to further enhance the FDTD method for current and future applications.
A Finite Difference informed Random Walk solver for simulating radiation defect evolution in polycrystalline structures with strongly inhomogeneous diffusivity
Diffusivity of species and defects on grain boundaries is usually several orders of magnitude larger than that inside grains. Such strongly inhomogeneous diffusivity requires prohibitively high computational demands for modeling microstructural evolution. Here, this paper presents a highly-efficient numerical solver, combining the Finite Difference method and Random Walk model, designed for accurately modeling strongly inhomogeneous diffusion within polycrystalline structures. The proposed solver, termed Finite Difference informed Random Walk (FDiRW), integrates a customized Finite Difference (cFD) scheme tailored for fast diffusion along thin grain boundaries represented by a single-layer of nodes. Numerical experiments demonstrate that the FDiRW solver achieves an impressive efficiency gain of 1560x compared to traditional Finite Difference methods while maintaining accuracy, making it feasible for personal computer machines to handle diffusional systems with strongly inhomogeneous diffusivity across static polycrystalline microstructures. The model has been successfully applied to simulate radiation defect evolution, showcasing its scalability to engineering scales in both length and time dimensions.
Deep Learning-Assisted Real-Time Forward Modeling of Electromagnetic Logging in Complex Formations
Higher dimensional (i.e., 2-D and 3-D) modeling is indispensable to correctly evaluate the responses of electromagnetic logging tools in complex formation environments. However, limited by the high computational cost of rigorous modeling such as finite difference method and the finite element method, the real-time applications in the well logging industry primarily rely on the 1-D forward solver, which would result in erroneous formation evaluation for complex scenarios. As a result, aiming at realizing fast modeling for electromagnetic logging tools in complex formations, this paper proposes a general framework assisted by deep neural networks (DNNs). The framework consists of three modules: earth model classification, parameter extraction, and surrogate construction. Separate DNNs are trained and tested for different modules. The accuracy and efficiency of the DNN assisted fast modeling are validated by several experiments. Here, this study finds that the fast modeling assisted by DNNs is able to calculate the tool responses and reconstruct the subsurface formations in real-time.