Showing papers on "Finite difference published in 2017"

Finite Difference Methods in Heat Transfer

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

Computation of Dendrites Using a Phase Field Model

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.

Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations

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.

Hybrid finite difference/finite element immersed boundary method.

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.

Applications of Lie Groups to Difference Equations

Abstract: Preface Introduction Brief introduction to Lie group analysis of differential equations Preliminaries: Heuristic approach in examples Finite Differences and Transformation Groups in Space of Discrete Variables The Taylor group and finite-difference derivatives Difference analog of the Leibniz rule Invariant difference meshes Transformations preserving the geometric meaning of finite-difference derivatives Newton's group and Lagrange's formula Commutation properties and factorization of group operators on uniform difference meshes Finite-difference integration and prolongation of the mesh space to nonlocal variables Change of variables in the mesh space Invariance of Finite-Difference Models An invariance criterion for finite-difference equations on the difference mesh Symmetry preservation in difference modeling: Method of finite-difference invariants Examples of construction of difference models preserving the symmetry of the original continuous models Invariant Difference Models of Ordinary Differential Equations First-order invariant difference equations and lattices Invariant second-order difference equations and lattices Invariant Difference Models of Partial Differential Equations Symmetry preserving difference schemes for the nonlinear heat equation with a source Symmetry preserving difference schemes for the linear heat equation Invariant difference models for the Burgers equation Invariant difference model of the heat equation with heat flux relaxation Invariant difference model of the Korteweg-de Vries equation Invariant difference model of the nonlinear Shrodinger equation Combined Mathematical Models and Some Generalizations Second-order ordinary delay differential equations Partial delay differential equations Symmetry of differential-difference equations Lagrangian Formalism for Difference Equations Discrete representation of Euler's operator Criterion for the invariance of difference functionals Invariance of difference Euler equations Variation of difference functional and quasi-extremal equations Invariance of global extremal equations and properties of quasiextremal equations Conservation laws for difference equations Noether-type identities and difference analog of Noether's theorem Necessary and sufficient conditions for global extremal equations to be invariant Applications of Lagrangian formalism to second-order difference equations Moving mesh schemes for the nonlinear Shrodinger equation Hamiltonian Formalism for Difference Equations: Symmetries and First Integrals Discrete Legendre transform Variational statement of the difference Hamiltonian equations Symplecticity of difference Hamiltonian equations Invariance of the Hamiltonian action Difference Hamiltonian identity and Noether-type theorem for difference Hamiltonian equations Invariance of difference Hamiltonian equations Examples Discrete Representation of Ordinary Differential Equations with Symmetries The discrete representation of ODE as a series Three-point exact schemes for nonlinear ODE Bibliography Index

Numerical Calculation of the Transonic Flow Past a Swept Wing

Abstract: A numerical method is presented for analyzing the transonic potential flow past a lifting, swept wing. A finite difference approximation to the full potential equation is solved in a coordinate system which is nearly conformally mapped from the physical space in planes parallel to the symmetry plane, and reduces the wing surface to a portion of one boundary of the computational grid. A coordinate invariant, rotated difference scheme is used, and the difference equations are solved by relaxation. The method is capable of treating wings of arbitrary planform and dihedral, although approximations in treating the tips and vortex sheet make its accuracy suspect for wings of small aspect ratio. Comparisons of calculated results with experimental data are shown for examples of both conventional and supercritical transport wings. Agreement is good for both types, but it was found necessary to account for the displacement effect of the boundary layer for the supercritical wing, presumably because of its greater sensitivity to changes in effective geometry.

Bounding the Costs of Quantum Simulation of Many-Body Physics in Real Space

Abstract: We present a quantum algorithm for simulating the dynamics of a first-quantized Hamiltonian in real space based on the truncated Taylor series algorithm. We avoid the possibility of singularities by applying various cutoffs to the system and using a high-order finite difference approximation to the kinetic energy operator. We find that our algorithm can simulate $\eta$ interacting particles using a number of calculations of the pairwise interactions that scales, for a fixed spatial grid spacing, as $\tilde{O}(\eta^2)$, versus the $\tilde{O}(\eta^5)$ time required by previous methods (assuming the number of orbitals is proportional to $\eta$), and scales super-polynomially better with the error tolerance than algorithms based on the Lie-Trotter-Suzuki product formula. Finally, we analyze discretization errors that arise from the spatial grid and show that under some circumstances these errors can remove the exponential speedups typically afforded by quantum simulation.

Finite Difference/Finite Element Methods for Distributed-Order Time Fractional Diffusion Equations

Abstract: In this paper, a class of distributed-order time fractional diffusion equations (DOFDEs) on bounded domains is considered. By L1 method in temporal direction, a semi-discrete variational formulation of DOFDEs is obtained firstly. The stability and convergence of this semi-discrete scheme are discussed, and the corresponding fully discrete finite element scheme is investigated. To improve the convergence rate in time, the weighted and shifted Grunwald difference method is used. By this method, another finite element scheme for DOFDEs is obtained, and the corresponding stability and convergence are considered. And then, as a supplement, a higher order finite difference scheme of the Caputo fractional derivative is developed. By this scheme, an approach is suggested to improve the time convergence rate of our methods. Finally, some numerical examples are given for verification of our theoretical analysis.

Detailed numerical solution of pore volume and surface diffusion model in adsorption systems

Abstract: In this work, the numerical solution of pore volume and surface diffusion model (PVSDM) was developed and presented in detail. The finite difference approximations method was employed to solve the partial differential equations of the diffusional model. The experimental adsorption of Malachite Green dye (MG) on bentonite clay was selected as case study. The equilibrium data were obtained from batch systems and, the Redlich–Peterson isotherm was suitable to represent the results. Due to non-linearity of the isotherm, the non-linear least squares technique was used to estimate the diffusional parameters. The Biot number has shown that the adsorption was simultaneously controlled by external mass transfer and intraparticle diffusion. Thus, both internal resistances (pore volume and surface) must be considered. The MG concentrations as a function of time decreases and the amount of MG mass adsorbed on bentonite clay as a function of the radial position increases until reaching equilibrium. The experimental concentration decay curve of MG was properly represented by PVDSM model. Further, the amount of MG mass adsorbed at higher radial positions was larger than at lower radial positions, within the PVSDM boundaries (particle boundaries), indicating that the method of finite difference approximations was appropriate for the numerical solution of PVSDM model.

Extended Algorithms for Approximating Variable Order Fractional Derivatives with Applications

Abstract: This paper proposes accurate and robust algorithms for approximating variable order fractional derivatives of arbitrary order. The proposed schemes are based on finite difference approximations. We compare the performance of algorithms by introducing a new formulation of experimental convergence order. Two initial value problems are considered and solved by means of the proposed methods. Numerical results are provided justifying the usefulness of the proposed methods.

A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

Abstract: We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in $\mathbb{R}^d$. The novelty of the method is in the approximation of the Laplace--Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace--Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computa...

Study on a Poisson's Equation Solver Based On Deep Learning Technique

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.

Numerical analysis of strongly nonlinear PDEs

Abstract: We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.

Discrete Exterior Calculus (DEC) for the Surface Navier-Stokes Equation

Abstract: We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on staggered grids in flat space for which we demonstrate second order convergence. We compare computational results with a vorticity-stream function approach for surfaces with genus \(g(\mathcal{S}) = 0\) and demonstrate the interplay between topology, geometry and flow properties. Our discretization also allows to handle harmonic vector fields, which we demonstrate on a torus.

Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid

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.

Regional Coastal Processes Numerical Modeling System: Report 1, Rcpwave-a Linear Wave Propagation Model for Engineering Use

Abstract: : The numerical model documented here, RCPWAVE, can be used to solve wave propagation problems over arbitrary bathymetry. The governing equations solved in the model are the mild slope equation for linear, monochromatic waves, and the equation specifying irrotationality of the wave phase function gradient. Finite difference approximations of these equations are solved to predict wave propagation outside the surf zone. Inside the breaker zone, an empirical method is used to predict wave transformation. This method is based on a hydraulic jump representation of the entire surf zone. The model is verified using laboratory and field data. A user's manual section is provided to aid potential users. This documentation describes job control language files, job submission procedures, sample input and output files, and execution costs.

Study on a Poisson's equation solver based on deep learning technique

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.

Transient thermal behavior of radial fins of rectangular, triangular and hyperbolic profiles with temperature-dependent properties using DTM-FDM

Abstract: This work focuses on transient thermal behavior of radial fins of rectangular, triangular and hyperbolic profiles with temperature-dependent properties. A hybrid numerical algorithm which combines differential transformation (DTM) and finite difference (FDM) methods is utilized to theoretically study the present problem. DTM and FDM are applied to the time and space domains of the problem, respectively. The accuracy of this method solution is checked against the numerical solution. Then, the effects of some applicable parameters were studied comparatively. Since a broad range of governing parameters are investigated, the results could be useful in a number of industrial and engineering applications.

An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation

Abstract: In this paper we propose and analyze an energy stable numerical scheme for the Cahn-Hilliard equation, with second order accuracy in time and the fourth order finite difference approximation in space. In particular, the truncation error for the long stencil fourth order finite difference approximation, over a uniform numerical grid with a periodic boundary condition, is analyzed, via the help of discrete Fourier analysis instead of the the standard Taylor expansion. This in turn results in a reduced regularity requirement for the test function. In the temporal approximation, we apply a second order BDF stencil, combined with a second order extrapolation formula applied to the concave diffusion term, as well as a second order artificial Douglas-Dupont regularization term, for the sake of energy stability. As a result, the unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the $\ell^\infty (0,T; \ell^2) \cap \ell^2 (0,T; H_h^2)$ norm. A few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

On the Decay Rates of Porous Elastic Systems

Abstract: In this paper we analyze the porous elastic system. We show that viscoelasticity is not strong enough to make the solutions decay in an exponential way, independently of any relationship between the coefficients of wave propagation speed. However, it decays polynomially with optimal rate. When the porous damping is coupled with microtemperatures, we give an explicit characterization on the decay rate that can be exponential or polynomial type, depending on the relation between the coefficients of wave propagation speed. Numerical experiments using finite differences are given to confirm our analytical results. It is worth noting that the result obtained here is different from all existing in the literature for porous elastic materials, where the sum of the two slow decay processes determine a process that decay exponentially.