Traditional Finite Difference Method Research Articles

The Modified Eulerian-Lagrangian Formulation for Cauchy Boundary Condition Under Dispersion Dominated Flow Regimes: A Novel Numerical Approach and its Implication on Radioactive Nuclide Migration or Solute Transport in the Subsurface Environment

The present study introduces a novel numerical approach for solving dispersion dominated problems with Cauchy boundary condition in an Eulerian-Lagrangian scheme. The study reveals the incapability of traditional Neuman approach to address the dispersion dominated problems with Cauchy boundary condition, even though it can produce reliable solution in the advection dominated regime. Also, the proposed numerical approach is applied to a real field problem of radioactive contaminant migration from radioactive waste repository which is a major current waste management issue. The performance of the proposed numerical approach is evaluated by comparing the results with numerical solutions of traditional FDM (Finite Difference Method), Neuman approach, and the analytical solution. The results show that the proposed numerical approach yields better and reliable solution for dispersion dominated regime, specifically for Peclet Numbers of less than 0.1. The proposed numerical approach is validated by applying to a real field problem of radioactive contaminant migration from radioactive waste repository of varying Peclet Number from 0.003 to 34.5. The numerical results of Neuman approach overestimates the concentration value with an order of 100 than the proposed approach during the assessment of radioactive contaminant transport from nuclear waste repository. The overestimation of concentration value could be due to the assumption that dispersion is negligible. Also our application problem confirms the existence of real field situation with advection dominated condition and dispersion dominated condition simultaneously as well as the significance or advantage of the proposed approach in the real field problem.

Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods – Part II: Numerical solution to the backward facing step flow

The lattice Boltzmann method (LBM) and traditional finite difference methods have separate strengths when solving the incompressible Navier–Stokes equations. The LBM is an explicit method with a highly local computational nature that uses floating-point operations that involve only local data and thereby enables easy cache optimization and parallelization. However, because the LBM is an explicit method, smaller grid spacing requires smaller numerical time steps during both transient and steady state computations. Traditional implicit finite difference methods can take larger time steps as they are not limited by the CFL condition, but only by the need for time accuracy during transient computations. To take advantage of the strengths of both methods, a multiple solver, multiple grid block approach was implemented and validated for the 2-D Burgers’ equation in Part I of this work. Part II implements the multiple solver, multiple grid block approach for the 2-D backward step flow problem. The coupled LBM–VSM solver is found to be faster by a factor of 2.90 (2.87 and 2.93 for Re=150 and Re=500, respectively) on a single processor than the VSM for the 2-D backward step flow problem while maintaining similar accuracy.

A Hybrid Finite Difference Method for Pricing Two-Asset Double Barrier Options

The pricing of the two-asset double barrier option is modeled as an initial-boundary value problem of the two-dimensional Black-Scholes partial differential equation. We use the hybrid finite different method to solve the problem. The hybrid method is a combination of the Laplace transform and a finite difference method. It is more efficient than a traditional finite difference method to obtain a solution without a step-by-step process. The method is implemented on a computer. Two numerical examples are calculated to verify the performance of the hybrid method. In our numerical examples, the convergence rate of the method is approximately two. We conclude that the method is efficient for pricing two-asset barrier options.

Integral Equations of the Stefan Problem and Their Application in Modeling of Thawing Soil

The paper offers a new numerical method for solving the Stefan problem, based on the reduction of the initial boundary value problem for nonlinear integral equation equivalent to the minimum dimension. A feature of the method is that the time-varying function of phase transition is used as a major unknown function. This parameterization is an efficient computational technique enabling the considerably simplified method of solution. Computational qualities of this method are investigated by the example of solving the practical problems of thawing frozen soils under thermal action produced by various ways. The method advantages are most evident in solving problems in an unbounded domain, as unlike traditional finite difference method there is no need to truncate the system of equations with a large possibly infinite dimension. Despite the complexity to construct Green's function, in the case of variable properties of the environment there is a broad class of problems concerning the simulation of phase transitions. Their solution can be successfully found by described method of non-linear integral equations. The method also provides additional convenience in Richardson extrapolation and in solution of Stephen inverse problems, in particular, problems of front control of phase transition.

A novel variable-grid finite-difference method for fiber lasers

A novel finite-difference method with a variable discrete grid is proposed. The algorithm is applied to the traveling wave model for fiber lasers successfully. The simulation for the CW and the actively Q-switched output of Yb-doped fiber laser is compared with the reported results and the results simulated by a traditional constant-grid finite difference method. The comparisons prove that the new algorithm causes negligible adverse impacts, while the efficiency is largely improved. The novel algorithm presented in this paper can be optimized to be a promising method for solving other sorts of partial differential equations.

Multiscale simulation of thermal contact resistance in electronic packaging

A multiscale model of thermal contact resistance (TCR) between two rough surfaces is developed. The lattice Boltzmann (LB) method and the traditional finite difference (FD) method are coupled to calculate the heat transfer between two rough surfaces. The LB method and the FD method are, respectively, applied to two different regions with different meshes (fine meshes and macro meshes). A coupling region for transmitting the boundary information between these two regions is established to ensure the continuity of physical parameters. In the multiscale model, the profile heights of actual surfaces are measured to construct the rough surface models, which have a higher accuracy than Gaussian surface. The elastic–plastic mechanics is applied to calculate the deformations of rough surfaces under pressure. A series of comparisons between both the numerical and experiments results are carried out to verify the validity of the multiscale model. The results show that the TCR is synthetically affected by several factors, such as surface roughness, surface flatness, pressure, microhardness, and thermal conductivity. In those parameters, the surface roughness, flatness, and thermal conductivity of materials exert great influences on the TCR under low-pressure condition. When the pressure increases, the pressure and microhardness play dominant roles in TCR.

An equivalent source model for the sound intensity in the vicinity of a high-performance military aircraft

The sound field of the F-22A Raptor has been measured extensively using arrays of microphones including an intensity probe to the sideline and rear of the aircraft. Recently, an equivalent source model (ESM) has been proposed that incorporates two arrays of monopole sources placed along the jet axis and their image sources to account for the hard ground. The monopole amplitudes in each case are described by a Rayleigh distribution, and one array includes a phase angle between the monopoles to produce correlated noise steered in a specific direction. The model closely replicates the measured SPL at the majority of measurement locations. However, agreement with energy-based quantities such as the vector acoustic intensity may not follow from agreement with the pressure measurements alone. Hence, in this work, the ESM’s ability to produce intensity estimates that match the measured F-22A intensity is evaluated. A new method of estimating acoustic intensity, known as the phase and amplitude gradient estimation (PAGE) method, is applied to the measured F-22A data. The PAGE method accurately estimates intensity in a larger bandwidth than the traditional finite difference method, expanding this discussion of the ESM's accuracy. In addition, sensitivity of ESM parameter selection, e.g., correlated source phase angle and peak amplitude location, on changes in the predicted intensity is examined. [Work sponsored by ONR.]

Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods—Part I: Formulation and application to the 2-D Burgers’ equation

The lattice Boltzmann method is being increasingly employed in the field of computational fluid dynamics due to its computational efficiency. Floating-point operations in the lattice Boltzmann method involve local data and therefore allow easy cache optimization and parallelization. Due to this, the cache-optimized lattice Boltzmann method has superior computational performance over traditional finite difference methods for solving unsteady flow problems. When solving steady flow problems, the explicit nature of the lattice Boltzmann discretization limits the time step size and therefore the efficiency of the lattice Boltzmann method for steady flows. To quantify the computational performance of the lattice Boltzmann method for steady flows, a comparison study between the lattice Boltzmann method (LBM) and the alternating direction implicit (ADI) method was performed using the 2-D steady Burgers’ equation. The comparison study showed that the LBM performs comparatively poor on high-resolution meshes due to smaller time step sizes, while on coarser meshes where the time step size is similar for both methods, the cache-optimized LBM performance is superior. Because flow domains can be discretized with multiblock grids consisting of coarse and fine grid blocks, the cache-optimized LBM can be applied on the coarse grid block while the traditional implicit methods are applied on the fine grid blocks. This paper finds the coupled cache-optimized lattice Boltzmann-ADI method to be faster by a factor of 4.5 over the traditional methods while maintaining similar accuracy.

Dirac quasinormal modes in spherically symmetric regular black holes

Using the WKB approximation, massless and massive Dirac quasinormal modes (QNMs) are studied in spherically symmetric regular spacetimes. We analyze the relationships between QNM frequencies and the parameters (angular momentum number $l$, magnetic monopole charge $\ensuremath{\beta}$, and the mass of the field $m$) and discuss the extreme charge of magnetic monopole ${\ensuremath{\beta}}_{e}$ for spherically symmetric regular black holes. Furthermore, we apply an expansion method to expand QNMs in inverse powers of $L=l+1/2$ and confirm good precision with $l>n$. Finally, we improve the traditional finite difference method to be available in the massive Dirac case and illuminate the dynamical evolution of the massive Dirac field.

Finite integration method for partial differential equations

A finite integration method is proposed in this paper to deal with partial differential equations in which the finite integration matrices of the first order are constructed by using both standard integral algorithm and radial basis functions interpolation respectively. These matrices of first order can directly be used to obtain finite integration matrices of higher order. Combining with the Laplace transform technique, the finite integration method is extended to solve time dependent partial differential equations. The accuracy of both the finite integration method and finite difference method are demonstrated with several examples. It has been observed that the finite integration method using either radial basis function or simple linear approximation gives a much higher degree of accuracy than the traditional finite difference method.

Inverse analysis of thermal conductivities in transient non-homogeneous and non-linear heat conductions using BEM based on complex variable differentiation method

This paper presents a new inverse analysis approach to sensitivity analysis and material property identification in transient non-homogeneous and non-linear heat conduction Boundary Element Method (BEM) analysis based on Complex Variable Differentiation Method (CVDM). In this approach, the material properties are taken as the optimization variables, and the sensitivity coefficients are computed by CVDM. The advantages of using CVDM are that the computation of partial derivatives of an implicit function is reduced to function calculation in a complex domain, and the parameter sensitivity coefficients can be determined in a more accurate way than the traditional Finite Difference Method (FDM). Based on BEM and CVDM in evaluation of the sensitivity matrix of heat flux, the parameter such as thermal conductivity can be accurately identified. Six numerical examples are given to demonstrate the potential of the proposed approach. The results indicate that the presented method is efficient for identifying the thermal conductivity with single or multiple parameters.

Shape optimization of a general bypass duct for tone noise reduction using continuous adjoint method

A novel continuous adjoint-based acoustic propagation method is proposed for low-noise turbofan duct design. A fan bypass duct tonal noise propagation model that is verified by comparison with an analytical solution of the modal radiation from a semi-infinite duct with the shear layer is enhanced with its continuous adjoint formulation, having been applied to design the bypass duct. First, this article presents the complete formulation of the time-dependent optimal design problem. Second, a continuous adjoint-based acoustic propagation method for two-dimensional bypass duct configurations is derived and presented. This article aims at describing the potential of the adjoint technique for aeroacoustic shape optimization. The implementation of the unsteady aeroacoustic adjoint method is validated by comparing the sensitivity derivative with that obtained by finite differences. Using a continuous adjoint formulation, the necessary aerodynamic gradient information is obtained with large computational savings over traditional finite-difference methods. The examples presented demonstrate that the combination of a continuous-adjoint algorithm with a noise prediction method can be an efficient design tool in the bypass duct noise design problem.

Time-Splitting Chebyshev-Spectral Method for the Schrödinger Equation in the Semiclassical Regime with Zero Far-Field Boundary Conditions

Semiclassical limit of Schrödinger equation with zero far-field boundary conditions is investigated by the time-splitting Chebyshev-spectral method. The numerical results of the position density and current density are presented. The time-splitting Chebyshev-spectral method is based on Strang splitting method in time coupled with Chebyshev-spectral approximation in space. Compared with the time-splitting Fourier-spectral method, the time-splitting Chebyshev-spectral method is unnecessary to treat the wave function as periodic and holds the smoothness of the wave function. For different initial conditions and potential (e.g. constant potential and harmonic potential), extensive numerical test examples in one-dimension are studied. The numerical results are in good agreement with the weak limit solutions. It shows that the time-splitting Chebyshev-spectral method is effective in capturing ?-oscillatory solutions of the Schrödinger equation with zero far-field boundary conditions. In addition, the time-splitting Chebyshev-spectral method surpasses the traditional finite difference method in the meshing strategy due to the exponentially high-order accuracy of Chebyshev-spectral method.

An efficient reverse time migration method using local nearly analytic discrete operator

In recent years, reverse time migration (RTM), the most powerful depth imaging method, has become the preferred imaging tool in many geologic settings because of its ability to handle complex velocity models including steeply dipping interfaces. Finite difference is one of the most popular numerical methods applied in RTM in the industry. However, it often encounters a serious issue of numerical dispersion, which is typically suppressed by reducing the propagation grid sizes, resulting in large computation and memory increment. Recently, a nearly analytic discrete operator has been developed to approximate the partial differential operators, from which many antidispersion schemes have been proposed, and are confirmed to be superior to conventional algorithms in suppressing numerical dispersion. We apply an optimal nearly analytic discrete (ONAD) method to RTM to improve its accuracy and performance. Numerical results show that ONAD can be used effectively in seismic modeling and migration based on the full wave equations. This method produces little numerical dispersion and requires much less computation and memory compared to the traditional finite-difference methods such as Lax-Wendroff correction method. The reverse time migration results of the 2D Marmousi model and the Sigsbee2B data set show that ONAD can improve the computational efficiency and maintain image quality by using large extrapolation grids.

Stable calculation of Gaussian-based RBF-FD stencils

Traditional finite difference (FD) methods are designed to be exact for low degree polynomials. They can be highly effective on Cartesian-type grids, but may fail for unstructured node layouts. Radial basis function-generated finite difference (RBF-FD) methods overcome this problem and, as a result, provide a much improved geometric flexibility. The calculation of RBF-FD weights involves a shape parameter ε. Small values of ε (corresponding to near-flat RBFs) often lead to particularly accurate RBF-FD formulas. However, the most straightforward way to calculate the weights (RBF-Direct) becomes then numerically highly ill-conditioned. In contrast, the present algorithm remains numerically stable all the way into the ε→0 limit. Like the RBF-QR algorithm, it uses the idea of finding a numerically well-conditioned basis function set in the same function space as is spanned by the ill-conditioned near-flat original Gaussian RBFs. By exploiting some properties of the incomplete gamma function, it transpires that the change of basis can be achieved without dealing with any infinite expansions. Its strengths and weaknesses compared with the Contour-Padé, RBF-RA, and RBF-QR algorithms are discussed.

A new 2D/3D accurate geophysical forward modelling technique: sub-domain Chebyshev spectral method

A new numerical approach, called the ?sub-domain Chebyshev spectral method?, has been developed to calculate differentiations in a curved coordinate system, which may be employed for 2D/3D geophysical forward modelling. The new method utilises non-linear transformations defined by the free-surface topography and subsurface interfaces and incorporates cubic-spline interpolations to convert the global domain into subdomains, and applies Chebyshev points in the model discretisation and computation of the spatial derivatives. Such effort makes the numerical differentiations have ?spectral accuracy? inside the subdomains whose boundaries match the free-surface topography and subsurface interfaces. 2D and 3D synthetic experiments have been performed with two geological models, both having different free-surface topographies and sub-surface interfaces. The computational errors of the new approach were compared with traditional finite-difference schemes, and the results show that the sub-domain Chebyshev spectral method is superior to traditional finite-difference method in its accuracy and applicable for all of the geophysical forward modelling problems.

A solution to linear elasticity using locally supported RBF collocation in a generalised finite-difference mode

Abstract This work presents a novel meshless numerical approach for the solution of linear elasticity problems, using locally supported RBF collocation. The Kansa (unsymmetric) RBF collocation method is used to form local collocation systems, which enforce the PDE governing and boundary operators. With the displacement values acting as the unknowns in the system, a sparse global system is formed. This global matrix is formed in a manner analogous to a finite difference method, with the displacement values at each internal node defined in terms of the displacements at other nodes within the local stencil. In contrast to traditional finite difference methods, here the RBF collocation assumes the role traditionally played by polynomial interpolants. The RBF collocation does itself satisfy the governing PDE operator at some collocation points, and therefore allows for a significantly more accurate reconstruction than is found from simple polynomial interpolants. In addition, the boundary operators (for applied displacement and applied surface traction) are enforced directly within the local RBF collocation systems, rather than being enforced at the global matrix. In contrast to traditional finite difference methods based on polynomial interpolation, the RBF collocation does not require a regular arrangement of nodes. Therefore, the proposed numerical method is directly applicable to unstructured datasets.

Effect of application area distributions of mass, force, and support on free and forced vibration behavior of viscoelastically supported plates

In real world systems, implementation areas of mass, force and supports have considerable quantity, unlike homogenized areas in numerical solutions. In this study, the effect of area distributions of non-structural additional mass, external force and supports on linear vibrating plate systems are analyzed. There are two key questions addressed in this study. First, the majority of the previous works cover the range of a fundamental mode, and do not answer the question of how results of the fundamental analyses can be generalized for the higher modes based on investigated parameters, particularly for the case of forced vibration. Second, how structural safety is threatened by focused idealizations has not been adequately studied before. Numerical results of this study are obtained by using the variational difference method (VDM), which is based on a variational procedure in conjunction with traditional finite difference method. VDM is applied for determining free vibration characteristics and steady-state responses to a sinusoidally varying force applied to a viscoelastically supported plate. Because the results of the investigated systems showed that the vibration characteristics had a high sensitivity to analyzed parameters (the amount of area distributions of non-structural additional mass, external force and supports), the idealization process of areas must be attentively employed. The study provides insight not only for a fundamental mode but also for higher modes. Obviously the results of this deterministic study have showed that theoretical importance shall be considered in practical applications.

Hybrid Spline Difference Method (HSDM) for Transient Heat Conduction

This study develops a high-accuracy numerical method named the hybrid spline difference method (HSDM), to replace the finite-difference method. Verification by transient heat transfer finds that the concept of the spline finite-difference method can greatly simplify the complexity of the traditional spline calculation, while having a solution procedure as in the finite-difference method. In addition, the hybrid spline technique can prevent the problem of selection of the optimal parameter, and increase the numerical accuracy to O(Δt2, Δx4, Δt2Δx2). This method features high accuracy and similarity with the concept of the finite-difference method, and could replace the traditional finite-difference method.

Numerical methods for a fluid mixture model

SUMMARY In this paper, we firstly apply generalized difference methods to solve a fluid mixture model. The model is usually used to describe the tissue deformations and contains a nonlinear hyperbolic equation and an elliptic equation. Most people have used finite difference methods for solving the elliptic equation and other schemes for solving the hyperbolic equation. It is well known that the accuracy of traditional finite difference method is not high. This may be a serious disadvantage in the fluid mixture model, which describes cell movements and tissue deformations. The numerical methods we propose to improve accuracy are based on generalized Galerkin methods and dual decomposition. By choosing suitable trial function space and test function space, our generalized upwind difference schemes exhibit second-order convergence in space for smooth problems and can eliminate numerical oscillations for discontinuous problems. Some numerical results are presented to demonstrate the advantages of our methods. Copyright © 2012 John Wiley & Sons, Ltd.