Hybrid Finite Difference Research Articles

Primal-dual algorithm for solving a convex image dejittering model with hybrid finite differences

Primal-dual algorithm for solving a convex image dejittering model with hybrid finite differences

A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers

In this paper, we study the numerical solutions of singularly perturbed convection-diffusion two-point BVP as well as one-dimensional parabolic convection-diffusion IBVP with discontinuous convection coefficient (positive throughout the domain) and source term. The analytical solutions of these kind of problems exhibit a boundary layer near x = 0 and a weak interior layer near x = ξ. We discretise the spatial domain by the piecewise-uniform Shishkin mesh and the temporal domain by a uniform mesh. To approximate the spatial derivatives, we apply the hybrid finite difference scheme. The implicit-Euler scheme is used for discretising the temporal derivative. For the time independent problem, we derive that the proposed hybrid scheme is e-uniformly convergent of almost second-order and for the time dependent problem, we also prove that the proposed scheme is e-uniformly convergent of almost second-order in space and first-order in time. To validate the theoretical estimates, some numerical results are presented.

A new monotonically stable discrete model for the solution of differential equations emanating from the evaporating raindrop

Most of the models of the dynamics of evaporating raindrop have been created using Ordinary Differential Equations. Some models assumed that evaporation is affected by air resistance that is negligible and proportional to the velocity of transition others assumed that the air resistance is significant and proportional to the square of the velocity. Because of the significant differences in the basic assumption used by various modelers, the solution of the resulting equations produced differed values There are yet no work to confirm these dependencies. In this work we obtain a class of hybrid finite difference schemes that can be used to obtain a reliable approximate solution to some of these differential equation models. The schemes were found to possess the same monotonic properties as the analytic solution.

A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers

In this paper, we study the numerical solutions of singularly perturbed convection-diffusion two-point BVP as well as one-dimensional parabolic convection-diffusion IBVP with discontinuous convection coefficient (positive throughout the domain) and source term. The analytical solutions of these kind of problems exhibit a boundary layer near x = 0 and a weak interior layer near x = ξ. We discretise the spatial domain by the piecewise-uniform Shishkin mesh and the temporal domain by a uniform mesh. To approximate the spatial derivatives, we apply the hybrid finite difference scheme. The implicit-Euler scheme is used for discretising the temporal derivative. For the time independent problem, we derive that the proposed hybrid scheme is ε-uniformly convergent of almost second-order and for the time dependent problem, we also prove that the proposed scheme is ε-uniformly convergent of almost second-order in space and first-order in time. To validate the theoretical estimates, some numerical results are presented.

A Hybrid Method Based on Leapfrog ADI-FDTD and FDTD for Solving Multiscale Transmission Line Network

This article proposes a novel hybrid finite-difference time-domain (FDTD) method to calculate the transient responses of the transmission line network based on the leapfrog alternating direction implicit FDTD (leapfrog ADI-FDTD) algorithm and the traditional FDTD method. The proposed hybrid method can be implemented by dividing the transmission line network into two parts that are the interconnects part in a printed circuit board solved by the leapfrog ADI-FDTD and the cable part solved by the traditional FDTD with different mesh sizes and the same time step, respectively. In addition, some boundary conditions should be introduced based on the modified nodal approach and Kirchhoff's law at the terminals of the transmission line network. The numerical results show that the proposed hybrid method is in good agreement with HSPICE. Especially, it is more efficient and applicative for the multiscale transmission line network problems due to the use of the implicit FDTD solver with unconditional stability compared with the traditional FDTD method.

Dispersion modeling in pore networks: A comparison of common pore-scale models and alternative approaches

Mass transfer in porous media resulting from dispersion occurs in a wide variety of applications such as water treatment, flow batteries, flow in aquifers, enhanced oil recovery, and packed-bed reactors. The underlying mechanisms of dispersion are the molecular diffusion superimposed on the advective transport induced by the fluid flow. Modeling dispersion in pore networks can be performed at a much lower computational cost compared to that in direct numerical simulations (DNS) such as finite element or the lattice Boltzmann methods, so it can be regarded as a suitable alternative provided its accuracy is sufficient. The most common approach to model dispersion in network models is based on the first-order upwind scheme, despite its known limitations in terms of accuracy for certain flow and transport regimes. In this study, three alternative pore-scale models for dispersion, which are more accurate than the existing ones, were derived and tested in pore network simulations. These models were adopted from the CFD literature and are based on a spatial discretization of the advection-diffusion equation using the hybrid and power-law finite difference schemes and the exact solution of the one-dimensional advection-diffusion equation. Finally, considering dispersion problems over arbitrary porous structures, consisting of stick-and-ball geometries, and different flow and mass transfer arrangements, the developed models were validated. Validation was carried-out through comparisons between results obtained with DNS, using a finite element solver, and those from pore network simulations. It is shown that under a wide range of dispersion regimes (up to the onset of the dispersion power-law regime), the relative error (with respect to DNS results) introduced by the power-law and exact solution-based models is consistently below 1%, whereas the use of the upwind scheme leads to >10% of relative error, depending on the dispersion regime. All the dispersion models developed in this study were implemented as part of the open-source network modeling package, OpenPNM.

Large-Eddy Simulations of Oil Droplet Aerosol Transport in the Marine Atmospheric Boundary Layer

In this study, a hybrid large-eddy simulation (LES) model is developed and applied to simulate the transport of oil droplet aerosols in wind over progressive water waves. The LES model employs a hybrid spectral and finite difference method for simulating the wind turbulence and a bounded finite-volume method for modeling the oil aerosol transport. Using a wave-following coordinate system and computational grid, the LES model captures the turbulent flow and oil aerosol fields in the region adjacent to the unsteady wave surface. A flat-surface case with prescribed roughness (representing a pure wind-sea) and a wavy-surface case with regular plane progressive 100 m long waves (representing long-crest long-wavelength ocean swells) are considered to illustrate the capability of the LES model and study the effects of long progressive waves on the transport of oil droplet aerosols with four different droplet diameters. The simulation results and statistical analysis reveal enhanced suspension of oil droplets in wind turbulence due to strong disturbance from the long progressive waves. The spatial distribution of the aerosol concentration also exhibits considerable streamwise variations that correlate with the phase of the long progressive waves.

3-D Implicit–Explicit Hybrid Finite Difference/Spectral Element/Finite Element Time Domain Method Without a Buffer Zone

A nonconformal hybrid finite difference time domain (FDTD)/finite element time domain (FETD) method was previously introduced, which implemented the hybridization through a buffer zone. Although this method has been demonstrated to be accurate and long-time stable, further efforts are still desirable to remove the buffer zone and to implement an implicit–explicit time integration from the perspective of practical applications. In this paper, a novel hybrid method is proposed, which not only successfully eliminates the necessity of the buffer zone without compromising the featured advantage (e.g., nonconformal mesh) but also effectively applies an implicit–explicit time integration scheme to improve the computational efficiency. Furthermore, the new method extends the hybridization to a broader level by incorporating the spectral element time domain (SETD) method based on the discontinuous Galerkin and domain decomposition techniques, resulting in a more general hybrid FDTD/SETD/FETD framework. The framework employs the explicit leapfrog time integration for the FDTD region while it employs the implicit Crank–Nicolson time integration for the FETD region. For the SETD region, either the implicit or explicit time integration can be employed, depending on the mesh sizes in it. When the implicit region becomes large, it can be further split into multiple subdomains to reduce computational complexity. Numerical examples are included to demonstrate the performance of the proposed hybrid method, which is accurate, long-time stable and more efficient than the hybrid method with a buffer zone.

A parameter uniform numerical scheme for singularly perturbed differential-difference equations with mixed shifts

In this paper, we consider a second-order singularly perturbed differential-difference equations with mixed delay and advance parameters. At first, we approximate the model problem by an upwind finite difference scheme on a Shishkin mesh. We know that the upwind scheme is stable and its solution is oscillation free, but it gives lower order of accuracy. So, to increase the convergence, we propose a hybrid finite difference scheme, in which we use the cubic spline difference method in the fine mesh regions and a midpoint upwind scheme in the coarse mesh regions. We establish a theoretical parameter uniform bound in the discrete maximum norm. To check the efficiency of the proposed methods, we consider test problems with delay, advance and the mixed parameters and the results are in agreement with our theoretical findings.

Evolution of the Diffusion-Induced Flow over a Disk Submerged in a Stratified Viscous Fluid

The paper presents the results of the mathematically simulated evolution of a 3D diffusion-induced flow over a disk (with diameter d and thickness H = 0.76 d) immersed in a linearly density-stratified incompressible viscous fluid (described by a system of the Navier–Stokes equations in the Boussinesq approximation). The disk rests at the level of the neutral buoyancy (coinciding with its symmetry axis z) and disturbs the homogeneity of the background diffusion flux in the fluid forming a complex system of slow currents (internal gravitational waves). Over time, two thin horizontal convection cells are formed at the upper and lower parts of the disk stretching parallel to the z axis and adjacent to the base cell with thickness d/2. This work is the first to analyze in detail the fundamental mechanism for the formation of each new half-wave near the vertical axis x (passing through the center of the disk) during half the buoyancy period of the fluid Tb. This mechanism is based on gravitational instability. The emergence of this instability is first detected at 0.473Tb at a height of 3.9 d above the center of the disk. The same mechanism is also implemented over the place where the body moves in the horizontal direction. The 3D vortex structure of the flow is visualized by constructing the isosurfaces of the imaginary part of the complex-conjugate eigenvalues of the velocity gradient tensor. In mathematical simulation, we employed a numerical method SMIF that has proved itself over three decades with an explicit hybrid finite-difference scheme for the approximation of the convective terms of the equations (second-order approximation, monotonicity).

A parameter-uniform hybrid finite difference scheme for singularly perturbed system of parabolic convection-diffusion problems

ABSTRACT We present the hybrid finite difference scheme for singularly perturbed system of parabolic convection-diffusion problems exhibiting overlapping boundary layers. We discretize the time derivative by the backward-Euler method and the spatial derivatives is discretized by the hybrid difference scheme on Shishkin mesh. We have shown that the presented numerical scheme is parameter-uniform convergent of first-order in temporal variable and almost second-order in spatial variable. Numerical experiments supporting the theoretical results are presented.

Performance Evaluation of Multiple Groundwater Flow and Nitrate Mass Transport Numerical Models

Benchmarking of different numerical models simulating groundwater flow and contaminant mass transport is the aim of the present study, in order to determine criteria for the selection of numerical model(s) that could be better tailored to the needs of a specific region. This analysis aims at evaluating the performance of a finite difference-based numerical model (MODFLOW-ΜΤ3DMS), a finite element-based numerical model (FEFLOW), and a hybrid finite element-finite difference coupling numerical model (Princeton Transport Code-PTC), all developed to simulate groundwater flow and nitrate mass transport in an alluvial aquifer. The evaluation of the models’ performance is assessed based on statistical measures and graphical performance analysis of the model point predictions to the observed values. The outcome of the analysis showed that among the three groundwater simulation models, FEFLOW algorithm exhibited the best performance in simulating both groundwater level and nitrate mass distribution. All simulation algorithms were found to offer different advantages, so in principle the selection of the appropriate model(s) should be done in accordance with the problem’s characteristics and/ or in a complementary way in order to achieve accurate representation of the aquifer system and thus optimal groundwater resources management. Even though the selection of the most suitable groundwater simulation algorithm is case-oriented, however, fractional gross error (FGE) was proven to be a reliable indicator that could be used by modelers to select the most suitable groundwater algorithm based on the available groundwater data.

Imaging the Optical Fields of Functionalized Silver Nanowires through Molecular TERS.

We image 4-mercaptobenzonitrile-functionalized silver nanowires (∼20 nm diameter) through tip-enhanced Raman scattering (TERS). The enhanced local optical field-molecular interactions that govern the recorded hyperspectral TERS images are dissected through hybrid finite-difference time-domain density functional theory simulations. Our forward simulations illustrate that the recorded spatiospectral profiles of the chemically functionalized nanowires may be reproduced by accounting for the interaction between orientationally averaged molecular polarizability derivative tensors and enhanced incident/scattered local fields polarized along the tip axis. In effect, we directly map the enhanced optical fields of the nanowire in real space through TERS. The simultaneously recorded atomic force microscopy (AFM) images allow a direct comparison between our attainable spatial resolution in topographic (13 nm) and TERS (5 nm) imaging measurements performed under ambient conditions. Overall, our described protocol enables local electric field imaging with few nm precision through molecular TERS, and it is therefore generally applicable to a variety of plasmonic nanostructures.

Lightning-Induced Voltages on a Distribution Line With Surge Arresters Using a Hybrid FDTD–SPICE Method

Installing lightning arresters is an effective way to protect distribution lines from lightning-induced voltages. Spacing effectiveness of lightning arresters is one of the main concerns in the industry. Different separation distances of arresters have been proposed in the literature. In these studies, the simulations are made with simplified circuit models, which are not sufficient to support their conclusions. This paper presents a hybrid finite-difference time-domain (FDTD) SPICE method for analyzing lightning-induced overvoltages on a distribution line with arresters being installed. The FDTD method gives a full-wave solution of the induced voltages. To handle nonlinear lightning arresters, this FDTD method is integrated with a circuit solver SPICE. A diode-based model for the arrester is proposed using a SPICE subcircuit and is coupled to the FDTD domain as a connecting port. For the verification purpose, the method is compared with the results obtained in the rocket-triggered lightning experiment, in which no arresters were installed. Lightning-induced overvoltages on the distribution line with different arrester spacing arrangements are analyzed finally. Efficient arrester spacing under both first and subsequent strokes is investigated.

An accelerated nonlocal Poisson‐Boltzmann equation solver for electrostatics of biomolecule

The nonlocal modified Poisson-Boltzmann equation (NMPBE) is one important variant of a commonly used dielectric continuum model, the Poisson-Boltzmann equation (PBE), for computing electrostatics of biomolecules. In this paper, an accelerated NMPBE solver is constructed by finite element and finite difference hybrid techniques. It is then programmed as a software package for computing electrostatic solvation and binding free energies for a protein in a symmetric 1:1 ionic solvent. Numerical results validate the new solver and its numerical stability. They also demonstrate that the new solver has much better performance than the corresponding finite element solver in terms of computer CPU time. Furthermore, they show that the binding free energies produced by NMPBE can match chemical experiment data better than those by PBE.

Parameter-uniform fractional step hybrid numerical scheme for 2D singularly perturbed parabolic convection–diffusion problems

This article focuses on developing and analyzing an efficient numerical scheme for solving two-dimensional singularly perturbed parabolic convection–diffusion initial-boundary value problems exhibiting a regular boundary layer. For approximating the time derivative, we use the Peaceman–Rachford alternating direction implicit method on uniform mesh and for the spatial discretization, a hybrid finite difference scheme is proposed on a special rectangular mesh which is tensor-product of piecewise-uniform Shishkin meshes in the spatial directions. We prove that the numerical scheme converges uniformly with respect to the perturbation parameter $$\varepsilon $$ and also attains almost second-order spatial accuracy in the discrete supremum norm. Finally, numerical results are presented to validate the theoretical results. In addition to this, numerical experiments are conducted to demonstrate the effect of the time-dependent boundary conditions in the order of convergence numerically by introducing the classical evaluation of the boundary data; and also the improvement in the spatial order of accuracy of the present method by considering the Bakhvalov–Shishkin mesh in the spatial directions.

Higher-order layer-adapted finite difference schemes for singularly perturbed Robin problems

In this paper, we construct the midpoint upwind scheme on the Shishkin mesh and on the Bakhvalov-Shishkin mesh respectively for solving the singularly perturbed Robin boundary value problem. The central divided difference is used to discretize the first derivative in the Robin boundary condition to achieve the higher-order uniform convergence. The elaborate ε-uniform pointwise error estimates O(N-1 ln N) for 1≤i≤pmN and O(N-2) for pmN <i<N with pm = 1/(4e)+1/4 on the Shishkin mesh and O(N-1) for 1≤i≤N/2 and O(N-2) N/2 <i<N on the Bakhvalov-Shishkin mesh are proved. We also construct the hybrid finite difference scheme that combines the midpoint upwind scheme on the coarse part with the central difference scheme on the fine part on the Shishkin mesh, and prove a better uniform convergence of orders O(N-2 ln2 N) for 1≤i≤phN and O(N-2) for phN <i<N with ph = 1/(2e). Finally, a numerical experiment illustrates that these error estimates are sharp and the convergence is uniform with respect to the perturbation parameter.

Hybrid finite difference\u2013finite element method to incorporate topography and bathymetry for two-dimensional magnetotelluric modeling

In this research, a new numerical method, called the hybrid finite difference–finite element (hybrid FD–FE) method, is developed to solve 2-D magnetotelluric modeling by taking advantage of both the finite difference (FD) and finite element (FE) methods. With the hybrid FD–FE method, the model is first discretized as rectangular blocks and separated into two zones: the FD and FE zones. The FD zone is set for the subregions where topography or bathymetry does not appear. The FD approximation, which is fast, accurate and requires less memory resources, is then applied. For the FE zones where topography or bathymetry exists, the rectangular blocks are transformed into quadrilateral elements to handle the topography or bathymetry appropriately. Then, the FE approximation with quadrilateral elements, which is more accurate for topography or bathymetry zones, is applied. The system of equations for the hybrid FD–FE method is then formed according to the FD and FE schemes. The obtained system is a combination of the FD and FE equations. Three numerical methods are applied to test models with and without topography and bathymetry. The accuracy and efficiency in terms of errors, computational time and memory storage are presented, compared and discussed. The numerical experiments indicate that the FD scheme has a shorter computational time than the other schemes when modeling without topography and retains accuracy equivalent to that of the FE method, whereas FE is more practical when modeling with topography and bathymetry. However, our proposed hybrid FD–FE method is efficient in both situations. Without topography or bathymetry, its efficiency and accuracy approach those of the FD scheme. With topography and bathymetry, the hybrid FD–FE method is as accurate as FE, but its speed is slightly slower than that of FD. In terms of memory storage, the hybrid FD–FE method consumes slightly more storage than the FD method. This hybrid FD–FE method can be further extended and implemented for 3-D magnetotelluric modeling for more efficient computation.

An ϵ-uniform hybrid numerical scheme for a singularly perturbed degenerate parabolic convection–diffusion problem

ABSTRACTIn this paper, we study the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of the problem exhibits a parabolic boundary layer in the neighbourhood of x=0. First, we use the backward-Euler finite difference scheme to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problem, we apply the hybrid finite difference scheme, which is a combination of central difference scheme and midpoint upwind scheme on piecewise uniform Shishkin mesh. We derive the error estimates, which show that the proposed hybrid scheme is ϵ-uniform convergent of almost second-order (up to a logarithmic factor) in space and first-order in time. Some numerical results have been carried out to validate the theoretical results.

Verification Modal Summation Technique for Synthetic and Observation Seismogram for Pidie Jaya Earthquake M6.5

Neo-Deterministic Seismic Hazard Assessment (NDSHA) method is a seismic hazard assessment method that has an advantage on realistic physical simulation of the source, propagation, and geological-geophysical structure. This simulation is capable on generating the synthetics seismograms at the sites that being observed. At the regional NDSHA scale, calculation of the strong ground motion is based on 1D modal summation technique because it is more efficient in computation. In this article, we verify the result of synthetic seismogram calculations with the result of field observations when Pidie Jaya earthquake on 7 December 2016 occurred with the moment magnitude of M6.5. Those data were recorded by broadband seismometers installed by BMKG (Indonesian Agency for Meteorology, Climatology and Geophysics). The result of the synthetic seismogram calculations verifies that some stations well show the suitability with observation while some other stations show the discrepancies with observation results. Based on the results of the observation of some stations, evidently 1D modal summation technique method has been well verified for thin sediment region (near the pre-tertiary basement), but less suitable for thick sediment region. The reason is that the 1D modal summation technique excludes the amplification effect of seismic wave occurring within thick sediment region. So, another approach is needed, e.g., 2D finite difference hybrid method, which is a part of local scale NDSHA method.