Abstract: Finite Difference Methods in Heat Transfer presents a clear, step-by-step delineation of finite difference methods for solving engineering problems governed by ordinary and partial differential equations, with emphasis on heat transfer applications The finite difference techniques presented apply to the numerical solution of problems governed by similar differential equations encountered in many other fields Fundamental concepts are introduced in an easy-to-follow mannerRepresentative examples illustrate the application of a variety of powerful and widely used finite difference techniques The physical situations considered include the steady state and transient heat conduction, phase-change involving melting and solidification, steady and transient forced convection inside ducts, free convection over a flat plate, hyperbolic heat conduction, nonlinear diffusion, numerical grid generation techniques, and hybrid numerical-analytic solutions

Abstract: Summary We present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model, which is derived from the variation of a free-energy functional, consisting of a temperature dependent bulk potential and a conformational entropy with a gradient-dependent anisotropic coefficient. We introduce a novel Invariant Energy Quadratization approach to transform the free-energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations. Based on the reformulated equivalent governing system, we develop a first and a second order semi-discretized scheme in time for the system, in which all nonlinear terms are treated semi-explicitly. The resulting semi-discretized equations consist of a linear elliptic equation system at each time step, where the coefficient matrix operator is positive definite and thus, the semi-discrete system can be solved efficiently. We further prove that the proposed schemes are unconditionally energy stable. Convergence test together with 2D and 3D numerical simulations for dendritic crystal growth are presented after the semi-discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes. Copyright © 2016 John Wiley & Sons, Ltd.

Abstract: A phase field model is used to numerically simulate the solidification of a pure material. We employ it to compute growth into an undercooled liquid for a one-dimensional spherically symmetric geometry and a planar two-dimensional rectangular region. The phase field model equation are solved using finite difference techniques on a uniform mesh. For the growth of a sphere, the solutions from the phase field equations for sufficiently small interface widths are in good agreement with a numerical solution to the classical sharp interface model obtained using a Green's function approach. In two dimensions, we simulate dendritic growth of nickel with four-fold anisotropy and investigate the effect of the level of anisotropy on the growth of a dendrite. The quantitative behavior of the phase field model is evaluated for varying interface thickness and spatial and temporal resolution. We find quantitatively that the results depend on the interface thickness and with the simple numerical scheme employed it is not practical to do computations with an interface that is sufficiently thin for the numerical solution to accurately represent a sharp interface model. However, even with a relatively thick interface the results from the phase field model show many of the features of dendritic growth and they are in surprisingly good quantitative agreement with the Ivantsov solution and microscopic solvability theory.

Abstract: We consider numerical methods for solving the fractional-in-space Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. A standard fully discretized scheme for this equation is considered, namely, using the conventional second-order Crank–Nicolson scheme in time and the second-order central difference approach in space. For the resulting nonlinear scheme, we propose a nonlinear iteration algorithm, whose unique solvability and convergence can be proved. The nonlinear iteration can avoid inverting a dense matrix with only $$\mathcal {O}(N\log N)$$ computation complexity. One major contribution of this work is to show that the numerical solutions satisfy discrete maximum principle under reasonable time step constraint. Based on the maximum stability, the nonlinear energy stability for the fully discrete scheme is established, and the corresponding error estimates are investigated. Numerical experiments are performed to verify the theoretical results.

Abstract: The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach uses a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach.

Abstract: A novel meshless numerical scheme, based on the generalized finite difference method (GFDM), is proposed to accurately analyze the two–dimensional shallow water equations (SWEs). The SWEs are a hyperbolic system of first-order nonlinear partial differential equations and can be used to describe various problems in hydraulic and ocean engineering, so it is of great importance to develop an efficient and accurate numerical model to analyze the SWEs. According to split-coefficient matrix methods, the SWEs can be transformed to a characteristic form, which can easily present information of characteristic in the correct directions. The GFDM and the second-order Runge-Kutta method are adopted for spatial and temporal discretization of the characteristic form of the SWEs, respectively. The GFDM is one of the newly-developed domain-type meshless methods, so the time-consuming tasks of mesh generation and numerical quadrature can be truly avoided. To use the moving-least squares method of the GFDM, the spatial derivatives at every node can be expressed as linear combinations of nearby function values with different weighting coefficients. In order to properly cooperate with the split-coefficient matrix methods and the characteristic of the SWEs, a new way to determine the shape of star in the GFDM is proposed in this paper to capture the wave transmission. Numerical results and comparisons from several examples are provided to verify the merits of the proposed meshless scheme. Besides, the numerical results are compared with other solutions to validate the accuracy and the consistency of the proposed meshless numerical scheme.

Abstract: The present analysis concentrates to examine the influence of both thermal and solutal stratification on magneto-hydrodynamics (MHD) nanofluid flow along an exponentially stretching sheet. Moreover, simultaneous effects of mixed convection and viscous dissipation are also analyzed to determine the thermal conductivity within the restricted domain. Energy and concentration equation consist of two important slip mechanisms, namely: the Brownian motion of nanoparticles and the thermophoresis due to concentration difference. By the mean of compatible similarity transformed, a system of PDEs is converted into the system of nonlinear ODEs. The resulting nonlinear ODEs are successfully solved via the implicit finite difference method (FDM). Obtained numerical solutions are plotted for each profile for different and converging values of including parameters. To validate the results, numerical values of Nusselt number are compared with the existing literature for a particular case. Obtained results present the significant impact of each parameter on temperature and concentration. Nanofluid flow behaviour is also observed via velocity profile.

Abstract: The development of a new five-stages symmetric two-step method of fourteenth algebraic order with vanished phase-lag and its first, second, third and fourth derivatives is analyzed in this paper. More specifically: (1) we will present the development of the new method, (2) we will determine the local truncation error (LTE) of the new proposed method, (3) we will analyze the local truncation error based on the radial time independent Schrodinger equation, (4) we will study the stability and the interval of periodicity of the new proposed method based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, (5) we will test the efficiency of the new obtained method based on its application on the coupled differential equations arising from the Schrodinger equation.

Abstract: In this work, we investigated the feasibility of applying deep learning techniques to solve 2D Poisson's equation. A deep convolutional neural network is set up to predict the distribution of electric potential in 2D. With training data generated from a finite difference solver, the strong approximation capability of the deep convolutional neural network allows it to make correct prediction given information of the source and distribution of permittivity. Numerical experiments show that the predication error can reach below one percent, with a significant reduction in CPU time compared with the traditional solver based on finite difference methods.

Abstract: Anisotropic and attenuating properties of subsurface media cause amplitude loss and waveform distortion in seismic wave propagation, resulting in negative influence on seismic imaging. To correct the anisotropy effect and compensate amplitude attenuation, a compensated-amplitude vertical transverse isotropic (VTI) least-squares reverse time migration (LSRTM) method is adopted. In this method, the attenuation term of an attenuated acoustic wave equation is extended to a VTI quasi-differential wave equation, which takes care of effects from anisotropy and attenuation. The finite-difference method is used to solve the equation, in which attenuation terms are solved in the wavenumber domain, and other terms are solved in the space or wavenumber domain. Stable regularization operators are derived and introduced to the equations to eliminate severe numerical noise in high-frequency components during backward propagation. Meanwhile, a demigration operator, migration operator, and gradient formula for att...

Abstract: In recent years, non-Newtonian fluids have received much attention due to their numerous applications, such as plastic manufacture and extrusion of polymer fluids. They are more complex than Newtonian fluids because the relationship between shear stress and shear rate is nonlinear. One particular subclass of non-Newtonian fluids is the generalized Oldroyd-B fluid, which is modelled using terms involving multi-term time fractional diffusion and reaction. In this paper, we consider the application of the finite difference method for this class of novel multi-term time fractional viscoelastic non-Newtonian fluid models. An important contribution of the work is that the new model not only has a multi-term time derivative, of which the fractional order indices range from 0 to 2, but also possesses a special time fractional operator on the spatial derivative that is challenging to approximate. There appears to be no literature reported on the numerical solution of this type of equation. We derive two new different finite difference schemes to approximate the model. Then we establish the stability and convergence analysis of these schemes based on the discrete $H^1$ norm and prove that their accuracy is of $O(\tau+h^2)$ and $O(\tau^{\min\{3-\gamma_s,2-\alpha_q,2-\beta\}}+h^2)$, respectively. Finally, we verify our methods using two numerical examples and apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette flow of a generalized Oldroyd-B fluid model. Our methods are effective and can be extended to solve other non-Newtonian fluid models such as the generalized Maxwell fluid model, the generalized second grade fluid model and the generalized Burgers fluid model.

Abstract: In this work, we investigated the feasibility of applying deep learning techniques to solve Poisson's equation. A deep convolutional neural network is set up to predict the distribution of electric potential in 2D or 3D cases. With proper training data generated from a finite difference solver, the strong approximation capability of the deep convolutional neural network allows it to make correct prediction given information of the source and distribution of permittivity. With applications of L2 regularization, numerical experiments show that the predication error of 2D cases can reach below 1.5\% and the predication of 3D cases can reach below 3\%, with a significant reduction in CPU time compared with the traditional solver based on finite difference methods.

Abstract: In this paper, a fast explicit and unconditionally stable finite-difference time-domain (FDTD) method is developed, which does not require a partial solution of a global eigenvalue problem. In this method, a patch-based single-grid representation of the FDTD algorithm is developed to facilitate both theoretical analysis and efficient computation. This representation results in a natural decomposition of the curl–curl operator into a series of rank-1 matrices, each of which corresponds to one patch in a single grid. The relationship is then theoretically analyzed between the fine patches and unstable modes, based on which an accurate and fast algorithm is developed to find unstable modes from fine patches with a bounded error. These unstable modes are then upfront eradicated from the numerical system before performing an explicit time marching. The resultant simulation is absolutely stable for the given time step irrespective of how large it is, the accuracy of which is also ensured. In addition, both lossless and general lossy problems are addressed in the proposed method. The advantages of the proposed method are demonstrated over the conventional FDTD and the state-of-the-art explicit and unconditionally stable FDTD methods by numerical experiments.

Abstract: The traditional high-order staggered-grid finite-difference (SGFD) method has high-order accuracy in space, but only the second-order accuracy in time, which makes the traditional SGFD method suffer from a large temporal dispersion error during long-distance wave propagation This paper develops temporal fourth- and sixth-order and spatial arbitrary evenorder SGFD schemes to model isotropic elastic wave propagation The temporal high-order SGFD schemes have smaller temporal dispersion than the traditional temporal second-order scheme, and thus allow larger time steps to attain a similar accuracy The developed temporal high-order SGFD schemes are applied to simulate a quasi-stress–velocity wave equation (QWE) that is derived in the framework of a $k$ -space approach A split QWE (SQWE) is further developed, and numerical simulation of SQWE results in separated P (compressional)-wave and S (shear)-wave Theoretical computational cost analysis verifies that the numerical simulation of QWE using the temporal fourthand sixth-order SGFD schemes is more efficient than the numerical simulation of the traditional stress–velocity wave equation using the traditional temporal second-order SGFD scheme in 2-D In 3-D, the temporal fourth-order SGFD scheme still runs faster than the traditional temporal second-order scheme; however, the temporal sixth-order scheme is more efficient only when a longer stencil length than 12 is adopted Numerical examples confirm the correctness of the developed elastic wave modeling schemes

Abstract: Time-domain finite difference (FDTD) methods are popular tools for three-dimensional (3-D) room acoustics modeling, but numerical dispersion is an inherent problem that can place limitations on the usable bandwidth of a given scheme. Compact explicit 27-point schemes and “large-star” schemes with high-order spatial differences offer improvements to the simplest scheme, but are ultimately limited by their second-order accuracy in time. In this paper, we use modified equation methods to derive FDTD schemes with high orders of accuracy in both space and time, resulting in significant improvements in numerical dispersion as compared to the aforementioned schemes. In comparison to such schemes, the high-order accurate schemes presented in this paper use significantly less memory and fewer operations when low error tolerances in numerical phase velocities are critical, leading to higher usable bandwidths for auralization purposes. Simulation results are also presented, demonstrating improved approximations to modal frequencies of a shoe-box room and free-space propagation of a bandlimited pulse.