Conventional Finite Difference Scheme Research Articles

2-D acoustic equation prestack reverse-time migration based on optimized combined compact difference scheme

Abstract Reverse-time migration (RTM) is widely regarded as one of the most accurate migration methods available today. A crucial step in RTM involves extending seismic wavefields forward and backward. Compared to the conventional central finite difference (CFD) scheme, the combined compact difference (CCD) scheme offers several advantages, including a shorter difference operator and the suppression of numerical dispersion under coarse grids. These attributes conserve memory and enhance effectiveness while maintaining the same level of differential precision. In this article, we begin with the five-point eighth-order CCD scheme and utilize the least squares method and Lagrange multiplier method to optimize the difference coefficients. This optimization is guided by the concept of dispersion-relation-preserving (DRP). The result is the acquisition of an optimized combined compact difference (OCCD) scheme, further enhancing the ability to suppress numerical dispersion. We thoroughly compare and analyze dispersion relationships and stability conditions. In addition, we examine several crucial steps in the RTM of the second-order acoustic wave equation. These steps include absorption boundary conditions, boundary storage strategy, and Poynting vector imaging conditions. Finally, we apply both the CCD and OCCD schemes in the RTM of the layered model, graben model, and SEG/EAGE salt model. We compare these results with those obtained from CFD's RTM. Numerical findings demonstrate that, in contrast to the CFD scheme, the CCD scheme effectively suppresses numerical dispersion and enhances imaging accuracy. Moreover, the optimized OCCD scheme further improves the ability to suppress numerical dispersion and can obtain better imaging results, which is an effective reverse time migration method suitable for coarse grid conditions.

Applying the angular spectrum representation to calculate the optical force density generated in dielectrics by tightly focused laser beams

The Angular Spectrum Representation (ASR) is applied to describe semi-analytically the optical force densities acting on linear dielectric media when a quasi-monochromatic tightly focused Gaussian beam is applied. This method is seen to be inherently faster than conventional finite-difference schemes. Numerical simulations of the optical force densities were also performed and found to be in agreement with the literature, providing a complementary tool for the study of opto-mechanical effects in matter.

A hybrid Cartesian-meshless method for the simulation of thermal flows with complex immersed objects

In this study, a hybrid Cartesian-meshless method is first extended to deal with the thermal flows with complex immersed objects. The temperature and flow fields are governed by energy conservation equations and Navier–Stokes equations with the Boussinesq approximation, respectively. The governing equations are solved by a conventional finite difference scheme on a Cartesian grid and generalized finite difference (GFD) with singular value decomposition (SVD) approximation on meshless nodes, with second-order accuracy. The present thermal SVD–GFD method is applied to simulate the following seven numerical examples over a wide range of governing parameters, including that with the high Prandtl number: (1) forced convection around a circular cylinder; (2) mixed convection around a stationary circular cylinder in a lid-driven cavity; (3) mixed convection involving a moving boundary in a cavity with two rotating circular cylinders; (4) sedimentation of a cold circular particle in a long channel; (5) freely falling of a sphere in viscous fluid with thermal buoyancy; (6) sedimentation of a torus with thermal convection; and (7) flow over a heated circular cylinder. The excellent agreement between the published data and the present numerical results demonstrate the good capability of the thermal SVD–GFD method to simulate the thermal flows with complex immersed objects, especially those involving fluid–structure interaction and the high Prandtl number.

Illustrative Application of the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems to the Nordheim–Fuchs Reactor Dynamics/Safety Model

The application of the recently developed “nth-order comprehensive sensitivity analysis methodology for nonlinear systems” (abbreviated as “nth-CASAM-N”) has been previously illustrated on paradigm nonlinear space-dependent problems. To complement these illustrative applications, this work illustrates the application of the nth-CASAM-N to a paradigm nonlinear time-dependent model chosen from the field of reactor dynamics/safety, namely the well-known Nordheim–Fuchs model. This phenomenological model describes a short-time self-limiting power transient in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. This model is sufficiently complex to demonstrate all the important features of applying the nth-CASAM-N methodology yet admits exact closed-form solutions for the energy released in the transient, which is the most important system response. All of the expressions of the first- and second-level adjoint functions and, subsequently, the first- and second-order sensitivities of the released energy to the model’s parameters are obtained analytically in closed form. The principles underlying the application of the 3rd-CASAM-N methodology for the computation of the third-order sensitivities are demonstrated for both mixed and unmixed second-order sensitivities. For the Nordheim–Fuchs model, a single adjoint computation suffices to obtain the six 1st-order sensitivities, while two adjoint computations suffice to obtain all of the 36 second-order sensitivities (of which 21 are distinct). This illustrative example demonstrates that the number of (large-scale) adjoint computations increases at most linearly within the nth-CASAM-N methodology, as opposed to the exponential increase in the parameter-dimensional space which occurs when applying conventional statistical and/or finite difference schemes to compute higher-order sensitivities. For very large and complex models, the nth-CASAM-N is the only practical methodology for computing response sensitivities comprehensively and accurately, overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems.

Steric and Slippage Effects on Mass Transport by Using an Oscillatory Electroosmotic Flow of Power-Law Fluids.

In this paper, the combined effect of the fluid rheology, finite-sized ions, and slippage toward augmenting a non-reacting solute’s mass transport due to an oscillatory electroosmotic flow (OEOF) is determined. Bikerman’s model is used to include the finite-sized ions (steric effects) in the original Poisson-Boltzmann (PB) equation. The volume fraction of ions quantifies the steric effects in the modified Poisson-Boltzmann (MPB) equation to predict the electrical potential and the ion concentration close to the charged microchannel walls. The hydrodynamics is affected by slippage, in which the slip length was used as an index for wall hydrophobicity. A conventional finite difference scheme was used to solve the momentum and species transport equations in the lubrication limit together with the MPB equation. The results suggest that the combined slippage and steric effects promote the best conditions to enhance the mass transport of species in about 90% compared with no steric effect with proper choices of the Debye length, Navier length, steric factor, Womersley number, and the tidal displacement.

The Effect of Wave-Induced Current and Coastal Structure on Sediment Transport at the Zengwen River Mouth

The mechanisms that control estuarine sediment transport are complicated due to the interaction between riverine flows, tidal currents, waves, and wave-driven currents. In the past decade, severe seabed erosion and shoreline retreat along the sandy coast of western Taiwan have raised concerns regarding the sustainability of coastal structures. In this study, ADCPs (Acoustic Doppler Current Profiler) and turbidity meters were deployed at the mouth of the Zengwen river to obtain the time series and the spatial distribution of flow velocities and turbidity during the base flow and flood conditions. A nearshore circulation model, SHORECIRC, has been adapted into a hybrid finite-difference/finite-volume, TVD (Total Variation Diminishing)-type scheme and coupled with the wave-spectrum model Simulating Waves Nearshore (SWAN). Conventional finite-difference schemes often produce unphysical oscillations when modeling coastal processes with abrupt bathymetric changes at river mouths. In contrast, the TVD-type finite volume scheme allows for robust treatment of discontinuities through the shock-capturing mechanism. The model reproduces water levels, waves, currents observed at the mouth of the Zengwen River reasonably well. The simulated residual sediment transport patterns demonstrate that the transport process at the river mouth is dominated by the interaction of the bathymetry and wave-induced currents when the riverine discharge was kept in reservoirs. The offshore residual transport causes erosion at the northern part of the river mouth, and the onshore residual transport causes accretion in the ebb tidal shoals around the center of the river mouth. The simulated morphological evolution displays significant changes on shallower deltas. The location with significant sea bed changes is consistent with the spot in which severe erosion occurred in recent years. Further analysis of morphological evolution is also discussed to identify the role of coastal structures, for example, the extension of the newly constructed groins near the river mouth.

Mass transport by an oscillatory electroosmotic flow of power-law fluids in hydrophobic slit microchannels

In this work, we study the hydrodynamics, concentration field, and mass transport of species due to an oscillatory electroosmotic flow that obeys a power law. An additional aspect that is considered in the analysis corresponds to the effect of the slippage condition at the walls of the microchannel. The governing equations that describe the involved phenomena are the following: equation of Poisson–Boltzmann for the electrical potential in the electric double layer, the momentum equation, and the species transport equation. These equations were simplified with the aid of the lubrication theory and were numerically solved by using a conventional finite difference scheme. Our results suggest that, under the slippage effects, the best conditions can be promoted for the mass transport of species for different values of the Schmidt number and Womersley numbers less than unity, and even it is maximized up to two orders of magnitude when $${\text {Wo}}>1$$ . In the analysis, the cross-over phenomenon appears for the mass transport for different species and it is identified for both Newtonian and non-Newtonian fluids. For shear-thinning fluids with slippage at the microchannel walls, the cross-over phenomenon occurs, and the species with less diffusivity can be transported up to ten times faster in comparison with Newtonian fluids when the no-slip effect is considered.

Optimisation of the finite-difference scheme based on an improved PSO algorithm for elastic modelling

The finite-difference (FD) scheme is extensively applied in seismic modelling, imaging and inversion due to its advantages of large-scale parallel computing and programming. However, numerical dispersion caused by using a difference operator in substitution for the differential operator is non-negligible, which reduces the accuracy of the modelling and can lead to some misinterpretations. In addition, the computing resources required by the FD scheme is highly demanding when dealing with large models, which limits its applicability. In this paper, a new optimised FD scheme is proposed, which is based on an improved particle swarm optimisation (PSO) algorithm. We improve the conventional PSO algorithm by introducing strategies related to local learning and global learning, which contribute to accelerating the convergence rate and effectively avoiding getting trapped in local extrema. Then, the improved PSO algorithm is used to improve the conventional FD scheme. Dispersion analysis and numerical modelling demonstrate that the low-order optimised FD scheme can achieve higher accuracy than a high-order conventional operator. Compared with the conventional FD scheme and a FD scheme based on the Remez exchange algorithm, the optimised FD scheme based on the improved PSO algorithm can more efficiently suppress numerical dispersion and increase computational efficiency.

Lattice Boltzmann simulations of thermal flows beyond the Boussinesq and ideal-gas approximations.

In this work, the recent lattice Boltzmann model with self-tuning equation of state (EOS) [R. Huang et al., J. Comput. Phys. 392, 227 (2019)]JCTPAH0021-999110.1016/j.jcp.2019.04.044 is improved in three aspects to simulate the thermal flows beyond the Boussinesq and ideal-gas approximations. First, an improved scheme is proposed to eliminate the additional cubic terms of velocity, which can significantly improve the numerical accuracy. Second, a local scheme is proposed to calculate the density gradient instead of the conventional finite-difference scheme. Third, a scaling factor is introduced into the lattice sound speed, which can be adjusted to effectively enhance numerical stability. The thermal Couette flow of a nonattracting rigid-sphere fluid, which is described by the Carnahan-Starling EOS, is first simulated, and the better performance of the present improvements on the numerical accuracy and stability is demonstrated. As a further application, the turbulent Rayleigh-Bénard convection in a supercritical fluid slightly above its critical point, which is described by the van der Waals EOS, is successfully simulated by the present lattice Boltzmann model. The piston effect of the supercritical fluid is successfully captured, which induces a fast and homogeneous increase of the temperature in the bulk region, and the time evolution from the initiation of heating to the final turbulent state is analyzed in detail and divided into five stages.

Numerical modeling of elastic wave in frequency-domain by using staggered grid fourth-order finite-difference scheme

Simulation of elastic wave propagation is an important method for oil and gas exploration. Accuracy and efficiency of elastic wave simulation in complex geological environment are always the focus issue. In order to improve the accuracy and efficient in numerical modeling of elastic modeling, a staggered grid fourth-order finite-difference scheme of modeling elastic wave in frequency-domain is developed, which can provide stable numerical solution with fewer number of grid points per wavelength. The method is implemented on first-order velocity-stress equation and a parsimonious spatial staggered-grid with fourth-order approximation of the first-order derivative operator. Numerical tests show that the accuracy of the fourth-order staggered-grid stencil is superior to that of the mixed-grid and other conventional finite difference stencils, especially in terms of shear-wave phase velocity. Measures of mass averaging acceleration and optimization of finite difference coefficients are taken to improve the accuracy of numerical results. Meanwhile, the numerical accuracy of the finite difference scheme can be further improved by enlarging the mass averaging area at the price of expanding the bandwidth of the impedance matrix that results in the reduction of the number of grid points to 3 per shear wavelength and computer storage requirement in simulation of practical models. In our scheme, the phase velocities of compressional and shear wave are insensitive to Poisson's ratio that does not occur to conventional finite difference scheme in most cases, and also the elastic wave modeling can degenerate to acoustic case automatically when the medium is pure fluid or gas. Furthermore, the staggered grid scheme developed in this study is suitable for modeling waves propagating in media with coupling fluid-solid interfaces that are not resolved very well for previous finite difference method. Cited as : Ma, C., Gao, Y., Lu, C. Numerical modeling of elastic wave in frequency-domain by using staggered grid fourth-order fifinite-difference scheme. Advances in Geo-Energy Research, 2019, 3(4): 410-423, doi: 10.26804/ager.2019.04.08

Analysis of transient combustion with the use of contemporary CFD techniques

The paper introduces the results of the validation of two contemporary CFD techniques for numerical analysis of transient combustion phenomena. Dissipation-free numerical technique (CABARET) and pseudo spectral method are outlined and validated by using several benchmarks. Three basic test routines are suggested among which are: one-dimensional steady flame, one-dimensional and two-dimensional unsteady flames. It is shown that both analyzed numerical techniques possess a set of advantages including extremely low values of scheme dissipation and dispersion. This allows reproducing the evolution of non-steady multidimensional flows with high accuracy. The disadvantages of conventional finite-difference schemes and the limits of their applicability are discussed. A full description of the proposed test routines is provided; one can utilize them to carry out validation of the reactive CFD codes.

Modeling the wave propagation in viscoacoustic media: An efficient spectral approach in time and space domain

We present an efficient and accurate modeling approach for wave propagation in anelastic media, based on a fractional spatial differential operator. The problem is solved with the Fourier pseudo-spectral method in the spatial domain and the REM (rapid expansion method) in the time domain, which, unlike the finite-difference and pseudo-spectral methods, offers spectral accuracy. To show the accuracy of the scheme, an analytical solution in a homogeneous anelastic medium is computed and compared with the numerical solution. We present an example of wave propagation at a reservoir scale and show the efficiency of the algorithm against the conventional finite-difference scheme. The new method, being spectral in the time and space simultaneously, offers a highly accurate and efficient solution for wave propagation in attenuating media.

Complex Absorbing Potential Formalism Accounting for Open Boundary Conditions Within the Wigner Transport Equation

For the derivation of the Wigner Transport Equation an infinite computational domain is presumed. As a consequence, when considering a bounded computational domain as it is the case for numerical device modeling, several difficulties arise based on this assumption. Due to the inadequate inclusion of the boundary terms, the Wigner function inherently suffers from an artificial interference pattern. To address this aspect, a complex absorbing potential formalism, which is originally utilized for the numerical solution of the Schrodinger equation, is developed for the application onto the Wigner Transport Equation. As a result, the drift operator includes an additional term accounting for the finiteness of the computational domain reflecting open boundary conditions. The approach is discussed by means of a simple structured resonant tunneling diode and utilizing conventional finite difference schemes for a numerical solution of the Wigner Transport Equation. Additionally, the results are compared with a reference solution obtained by the Quantum Transmitting Boundary Method.

A finite-difference approach for solving pure quasi-P-wave equations in transversely isotropic and orthorhombic media

Starting from the dispersion relation and setting S-wave velocity along symmetry axes to zero, pseudoacoustic-wave equations have been proposed to describe the kinematics of acoustic wavefields in transversely isotropic (TI) and orthorhombic media. To date, the numerical stability of the pseudoacoustic-wave equations has been improved by developing coupled systems of wave equations; however, most simulations still suffer from S-wave artifacts that are the fundamental solutions of the fourth- and sixth-order partial differential equations. Pure quasi-P-wave equations accurately describe the traveltimes of P-waves in TI and orthorhombic media and are free of S-wave artifacts. However, it is difficult to directly solve the pure quasi-P-wave equations using conventional finite-difference schemes due to the presence of pseudo-differential operators. We approximated these pseudo-differential operators by algebraic expressions, whose coefficients can be determined by minimizing differences between the true and approximated values of the pseudo-differential operators in the wavenumber domain. The derived new coupled systems involve modified acoustic-wave equations and a Poisson’s equation that can be solved by conventional finite-difference stencils and fast Poisson’s solver. Several 2D and 3D numerical examples demonstrate that the simulations based on the new systems are free of S-wave artifacts and have correct kinematics of quasi-P-waves in TI and orthorhombic media.

Acoustic VTI Modeling Using an Optimal Time-Space Domain Finite-Difference Scheme

Finite-difference (FD) schemes have been used widely for solving wave equations in seismic exploration. However, the conventional FD schemes hardly guarantee high accuracy at both small and large wavenumbers. In this paper, we propose an optimal time-space domain FD scheme for acoustic vertical transversely isotropic (VTI) wave modeling. The optimal FD coefficients for the second-order spatial derivatives are derived by approaching the time-space domain dispersion relation of acoustic VTI wave equations with the combination of the Taylor-series expansion and the sampling interpolation. We perform numerical dispersion analyses and acoustic VTI modeling using the optimal time-space domain FD scheme. The numerical dispersion analyses show that the optimal FD scheme has smaller dispersion than the conventional FD scheme at large wavenumbers, and also preserves high accuracy at small wavenumbers. The acoustic VTI modeling examples also demonstrate that the optimal time-space domain FD scheme has greater accuracy compared with the conventional time-space domain FD scheme for the same modeling parameters.

A Comparative Study of Finite Element and Finite Difference Methods for Two-Dimensional Space-Fractional Advection-Dispersion Equation

AbstractThe paper makes a comparative study of the finite element method (FEM) and the finite difference method (FDM) for two-dimensional fractional advection-dispersion equation (FADE) which has recently been considered a promising tool in modeling non-Fickian solute transport in groundwater. Due to the non-local property of integro-differential operator of the space-fractional derivative, numerical solution of FADE is very challenging and little has been reported in literature, especially for high-dimensional case. In order to effectively apply the FEM and the FDM to the FADE on a rectangular domain, a backward-distance algorithm is presented to extend the triangular elements to generic polygon elements in the finite element analysis, and a variable-step vector Grünwald formula is proposed to improve the solution accuracy of the conventional finite difference scheme. Numerical investigation shows that the FEM compares favorably with the FDM in terms of accuracy and convergence rate whereas the latter enjoys less computational effort.

Lattice Boltzmann model for collisionless electrostatic drift wave turbulence obeying Charney–Hasegawa–Mima dynamics

A lattice Boltzmann method (LBM) approach to the Charney–Hasegawa–Mima (CHM) model for adiabatic drift wave turbulence in magnetised plasmas is implemented. The CHM-LBM model contains a barotropic equation of state for the potential, a force term including a cross-product analogous to the Coriolis force in quasigeostrophic models, and a density gradient source term. Expansion of the resulting lattice Boltzmann model equations leads to cold-ion fluid continuity and momentum equations, which resemble CHM dynamics under drift ordering. The resulting numerical solutions of standard test cases (monopole propagation, stable drift modes and decaying turbulence) are compared to results obtained by a conventional finite difference scheme that directly discretizes the CHM equation. The LB scheme resembles characteristic CHM dynamics apart from an additional shear in the density gradient direction. The occurring shear reduces with the drift ratio and is ascribed to the compressible limit of the underlying LBM.

Computation of singular solutions to the Helmholtz equation with high order accuracy

Solutions to elliptic PDEs, in particular to the Helmholtz equation, become singular near the boundary if the boundary data do not possess sufficient regularity. In that case, the convergence of standard numerical approximations may slow down or cease altogether. We propose a method that maintains a high order of grid convergence even in the presence of singularities. This is accomplished by an asymptotic expansion that removes the singularities up to several leading orders, and the remaining regularized part of the solution can then be computed on the grid with the expected accuracy.The computation on the grid is rendered by a compact finite difference scheme combined with the method of difference potentials. The scheme enables fourth order accuracy on a narrow 3×3 stencil: it uses only one unknown variable per grid node and requires only as many boundary conditions as needed for the underlying differential equation itself. The method of difference potentials enables treatment of non-conforming boundaries on regular structured grids with no deterioration of accuracy, while the computational complexity remains comparable to that of a conventional finite difference scheme on the same grid. The method of difference potentials can be considered a generalization of the method of Calderon's operators in PDE theory.In the paper, we provide a theoretical analysis of our combined methodology and demonstrate its numerical performance on a series of tests that involve Dirichlet and Neumann boundary data with various degrees of “non-regularity”: an actual jump discontinuity, a discontinuity in the first derivative, a discontinuity in the second derivative, etc. All computations are performed on a Cartesian grid, whereas the boundary of the domain is a circle, chosen as a simple but non-conforming shape. In all cases, the proposed methodology restores the design rate of grid convergence, which is fourth order, in spite of the singularities and regardless of the fact that the boundary is not aligned with the discretization grid. Moreover, as long as the location of the singularities is known and remains fixed, a broad spectrum of problems involving different boundary conditions and/or data on “smooth” segments of the boundary can be solved economically since the discrete counterparts of Calderon's projections need to be calculated only once and then can be applied to each individual formulation at very little additional cost.

Nonequilibrium scheme for computing the flux of the convection-diffusion equation in the framework of the lattice Boltzmann method.

In this paper, we propose a local nonequilibrium scheme for computing the flux of the convection-diffusion equation with a source term in the framework of the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). Both the Chapman-Enskog analysis and the numerical results show that, at the diffusive scaling, the present nonequilibrium scheme has a second-order convergence rate in space. A comparison between the nonequilibrium scheme and the conventional second-order central-difference scheme indicates that, although both schemes have a second-order convergence rate in space, the present nonequilibrium scheme is more accurate than the central-difference scheme. In addition, the flux computation rendered by the present scheme also preserves the parallel computation feature of the LBM, making the scheme more efficient than conventional finite-difference schemes in the study of large-scale problems. Finally, a comparison between the single-relaxation-time model and the MRT model is also conducted, and the results show that the MRT model is more accurate than the single-relaxation-time model, both in solving the convection-diffusion equation and in computing the flux.

A high-order hybrid finite difference–finite volume approach with application to inviscid compressible flow problems: A preliminary study

A class of hybrid finite difference–finite volume (FD–FV) operators is recently developed as building blocks to solve one dimensional hyperbolic conservation laws when the solutions are smooth. This method differs from conventional finite difference (FD) or finite volume (FV) schemes in that both nodal values and cell-averaged values are considered as dependent variables and they are evolved in time. Under this framework, the 1D FD–FV methods: (1) are numerically conservative for cell averages; (2) have straightforward extension to high-order accuracy; and (3) have superior spatial accuracy property compared to most conventional FD or FV methods.This work extends the FD–FV approach in two aspects. The first extension is a WENO-type stabilization to enhance the nonlinear stability of sample high-order 1D FD–FV operators. In particular, numerical results show that when the solutions are smooth, the optimal order of accuracy (fifth-order) is achieved by the stabilized fourth-order FD–FV method; and it is also capable to handle problems with strongly discontinuous solutions.The second part of the paper extends a second-order FD–FV method to two-dimensional smooth problems. Both Cartesian grids and unstructured (triangular) grids are considered. In multiple dimensions, there are different choices of the collocation points of the nodal values, and they lead to different FD–FV schemes. This work develops a node-centered FD–FV scheme and an edge-centered FD–FV scheme on each type of grids, and their numerical performance are assessed and compared by solving benchmark flow problems with smooth solutions. In particular, the numerical examples confirm that the superior spatial accuracy property of the 1D FD–FV operators carries to two space dimensions on Cartesian grids. The present work focuses on two space dimensions, but the methodology extends naturally to three-dimensional Cartesian grids and tetrahedral grids.