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Showing papers on "Finite difference published in 2019"


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04 Nov 2019
TL;DR: In this paper, the authors consider linear initial value problems, Sturm-Liouville problems and related inequalities in several independent variables, including difference inequalities and boundary value problems for linear systems and nonlinear systems.
Abstract: Preliminaries linear initial value problems miscellaneous difference equations difference inequalities qualitative properties of solutions of difference systems qualitative properties of solutions of higher order difference equations qualitative properties of solutions of neutral difference equations boundary value problems for linear systems boundary value problems for nonlinear systems miscellaneous properties of solutions of higher order linear difference equations boundary value problems for higher order difference equations Sturm-Liouville problems and related inequalities difference inequalities in several independent variables.

939 citations


Journal ArticleDOI
TL;DR: In this article, an initial-boundary value problem with a Caputo time derivative of fractional order α ∆ in(0,1) is considered, and a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the $L ∆ ∆ and $L_2$ norms is presented.
Abstract: An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the $L_\infty$ and $L_2$ norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.

137 citations


Journal ArticleDOI
TL;DR: The unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the $\ell^\infty (0,T; T; H_h^2)$ norm.

127 citations


Journal ArticleDOI
01 Jun 2019
TL;DR: The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role.
Abstract: In this paper we present and analyze finite difference numerical schemes for the Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both first and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that this numerical algorithm has a unique solution, such that the positivity is always preserved for the logarithmic arguments, i.e., the phase variable is always between −1 and 1, at a point-wise level. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of −1 and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. Such an analysis technique can also be applied to a second order numerical scheme in which the BDF temporal stencil is applied, the expansive term is updated by a second order Adams-Bashforth explicit extrapolation formula, and an artificial Douglas-Dupont regularization term is added to ensure the energy dissipativity. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, namely, the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes.

122 citations


Journal ArticleDOI
TL;DR: In this paper, a proper orthogonal decomposition (POD) is proposed to reduce the order of the classical Crank-Nicolson finite difference (CCNFD) model for the fractional-order parabolic-type sine-Gordon equations (FOPTSGEs).
Abstract: In this paper, by means of a proper orthogonal decomposition (POD) we mainly reduce the order of the classical Crank–Nicolson finite difference (CCNFD) model for the fractional-order parabolic-type sine-Gordon equations (FOPTSGEs). Toward this end, we will first review the CCNFD model for FOPTSGEs and the theoretical results (such as existence, stabilization, and convergence) of the CCNFD solutions. Then we establish an optimized Crank–Nicolson finite difference extrapolating (OCNFDE) model, including very few unknowns but holding the fully second-order accuracy for FOPTSGEs via POD. Next, by a matrix analysis we will discuss the existence, stabilization, and convergence of the OCNFDE solutions. Finally, we will use a numerical example to validate the validity of theoretical conclusions. Moreover, we show that the OCNFDE model is very valid for settling FOPTSGEs.

122 citations


Journal ArticleDOI
TL;DR: In this paper, the coupled Boussinesq equation which arises in the shallow water waves for two-layered fluid flow was investigated and the modified exp $$(-\varphi (\zeta ))-expansion function method was utilized in reaching the solutions to this equation such as the topological kink-type soliton and singular soliton solutions.
Abstract: The studies of the dynamic behaviors of nonlinear models arising in ocean engineering play a significant role in our daily activities. In this study, we investigate the coupled Boussinesq equation which arises in the shallow water waves for two-layered fluid flow. The modified exp $$(-\varphi (\zeta ))$$ -expansion function method is utilized in reaching the solutions to this equation such as the topological kink-type soliton and singular soliton solutions. The interesting 2D and 3D graphics of the obtained analytical solutions in this study are presented. Via one of the reported analytical solutions, the finite forward difference method is used in obtaining the approximate numerical and exact solutions to this equation. The Fourier–Von Neumann analysis is used in checking the stability of the used numerical method with the studied model. The $$L_{2}$$ and $$L_{\infty }$$ error norms are computed. We finally present a comprehensive conclusion to this study.

71 citations


Journal ArticleDOI
TL;DR: In this paper, two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation were presented, one based on the conventional second-order central difference quotient but with a cell-centered grid, and the other based on a regular grid but incorporated with summation by parts (SBP) operators.

63 citations


Journal ArticleDOI
TL;DR: This study aims at gaining a better understanding of the behavior of PHS+poly generated RBF-FD approximations near boundaries, illustrating it in 1-D, 2-D and 3-D.

60 citations


Journal ArticleDOI
24 Apr 2019
TL;DR: In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f (t) appearing inside the integral sign of the definition of the CF operator.
Abstract: In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f (t) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration.

58 citations


Journal ArticleDOI
TL;DR: The present study complements the previous results, providing an analytical insight into RBF-FD approximations augmented with polynomials, based on a closed-form expression for the interpolant, which reveals the mechanisms underlying these features, including the role of polynmials and RBFs in the interpolants, the approximation error, and the behavior of the cardinal functions near boundaries.
Abstract: Radial basis function-generated finite differences (RBF-FD) based on the combination of polyharmonic splines (PHS) with high degree polynomials have recently emerged as a powerful and robust numerical approach for the local interpolation and derivative approximation of functions over scattered node layouts. Among the key features, (i) high orders of accuracy can be achieved without the need of selecting a shape parameter or the issues related to numerical ill-conditioning, and (ii) the harmful edge effects associated to the use of high order polynomials (better known as Runge’s phenomenon) can be overcome by simply increasing the stencil size for a fixed polynomial degree. The present study complements our previous results, providing an analytical insight into RBF-FD approximations augmented with polynomials. It is based on a closed-form expression for the interpolant, which reveals the mechanisms underlying these features, including the role of polynomials and RBFs in the interpolant, the approximation error, and the behavior of the cardinal functions near boundaries. Numerical examples are included for illustration.

56 citations


Journal ArticleDOI
TL;DR: The finite element method is considered for the novel 2D multi-term time fractional mixed diffusion equation, which possesses the diffusion-wave and sub-diffusion terms simultaneously but also has a special time-space coupled derivative.

Journal ArticleDOI
TL;DR: In this article, a new finite element numerical method for the solution of partial differential equations on evolving domains is proposed. But the method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain.
Abstract: The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain. The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. This paper includes a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. Several numerical examples illustrate the theory and demonstrate the practical efficiency of the method.

Journal ArticleDOI
TL;DR: A wavelet method, based on Haar wavelets and finite difference scheme for two-dimensional time fractional reaction–subdiffusion equation, and performance is compared with other methods available in literature such as meshless-based methods and compact alternating direction implicit methods.
Abstract: In this study, we established a wavelet method, based on Haar wavelets and finite difference scheme for two-dimensional time fractional reaction–subdiffusion equation. First by a finite difference approach, time fractional derivative which is defined in Riemann–Liouville sense is discretized. After time discretization, spatial variables are expanded to truncated Haar wavelet series, by doing so a fully discrete scheme obtained whose solution gives wavelet coefficients in wavelet series. Using these wavelet coefficients approximate solution constructed consecutively. Feasibility and accuracy of the proposed method is shown on three test problems by measuring error in $$L_{\infty }$$ norm. Further performance of the method is compared with other methods available in literature such as meshless-based methods and compact alternating direction implicit methods.

Journal ArticleDOI
14 Aug 2019
TL;DR: In this paper, the authors analyzed the unsteady natural convection with the help of fractional approach and applied the finite difference approach coupled with Crank Nicolson method to investigate the numerical solutions of non-dimensional system of partial differential equations.
Abstract: In the current article, we analyzed the unsteady natural convection with the help of fractional approach. Firstly, the unsteady natural convection radiating flow in an open ended vertical channel beside the magnetic effects. We assumed the channel is stationary with non-uniform temperature. Secondly, we utilized a fractional calculus approach for the constitutive relationship of a fluid model. The modeled problem is transformed into nondimensional form via viable non-dimensional variables. In order to investigate the numerical solutions of non-dimensional system of partial differential equations finite difference approach coupled with Crank Nicolson method is developed and successfully applied. The beauty of Crank Nicolson finite difference scheme is, this scheme is unconditionally stable. A very careful survey of literate witnesses that this scheme has never been reported in the literary for fluid problems. The physical changes are discussed with the help of graphics. The expression for both velocity field and temperature distribution has been made via said scheme. A comprehensive discussion about the influence of various related dimensionless parameters upon the flow properties disclosed our work. It is observed that velocity field decreases as enhancing the magnetic field effects. Heat transfer enhanced as enhancing the nanoparticle volume fraction parameter. Velocity field and heat transfer shows the dominant behavior for the case of Cu-based nanofluid as compare to Al2O3 based nanofluid. Comparative study also included to show the accuracy of the proposed finite difference scheme. It is to be highlighted that the proposed scheme is very efficient and well-matched to investigate the solutions of modeled problem and can be extended to diversify problems of physical nature.

Journal ArticleDOI
Tao Wang1, Xuan Ye1, Zhanli Liu1, Dongyang Chu1, Zhuo Zhuang1 
TL;DR: The driving force of phase field evolution based on Mohr–Coulomb criterion for rock and other materials with shear frictional characteristics is derived and a three-dimensional explicit parallel phase field model is developed.
Abstract: The phase field method is a very effective method to simulate arbitrary crack propagation, branching, convergence and complex crack networks. However, most of the current phase-field models mainly focus on tensile fracture problems, which is not suitable for rock-like materials subjected to compression and shear loads. In this paper, we derive the driving force of phase field evolution based on Mohr–Coulomb criterion for rock and other materials with shear frictional characteristics and develop a three-dimensional explicit parallel phase field model. In spatial integration, the standard finite element method is used to discretize the displacement field and the phase field. For the time update, the explicit central difference scheme and the forward difference scheme are used to discretize the displacement field and the phase field respectively. These time integration methods are implemented in parallel, which can tackle the problem of the low computational efficiency of the phase field method to a certain extent. Then, three typical benchmark examples of dynamic crack propagation and branching are given to verify the correctness and efficiency of the explicit phase field model. At last, the failure processes of rock-like materials under quasi-static compression load are studied. The simulation results can well capture the compression-shear failure mode of rock-like materials.

Journal ArticleDOI
TL;DR: A novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds, which avoids the complexities of dealing with a manifold metric, while also avoiding the need to solve a PDE in the embedding space.
Abstract: In this paper, we propose a novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative approximations for the same are done directly on the tangent space, in a manner that mimics the procedure followed in volume-based meshfree GFDMs. As a result, the proposed method not only does not require a mesh, it also does not require an explicit reconstruction of the manifold. In contrast to some existing methods, it avoids the complexities of dealing with a manifold metric, while also avoiding the need to solve a PDE in the embedding space. A major advantage of this method is that all developments in usual volume-based numerical methods can be directly ported over to surfaces using this framework. We propose discretizations of the surface gradient operator, the surface Laplacian and surface Diffusion operators. Possibilities to deal with anisotropic and discontinuous surface properties with large jumps are also introduced, and a few practical applications are presented.

Journal ArticleDOI
TL;DR: A family of modified Patankar Runge–Kutta methods is constructed, which is conservative and unconditionally positivity-preserving, for production–destruction equations, and derive necessary and sufficient conditions to obtain second-order accuracy.
Abstract: In this paper, we construct a family of modified Patankar Runge–Kutta methods, which is conservative and unconditionally positivity-preserving, for production–destruction equations, and derive necessary and sufficient conditions to obtain second-order accuracy. This ordinary differential equation solver is then extended to solve a class of semi-discrete schemes for PDEs. Combining this time integration method with the positivity-preserving finite difference weighted essentially non-oscillatory (WENO) schemes, we successfully obtain a positivity-preserving WENO scheme for non-equilibrium flows. Various numerical tests are reported to demonstrate the effectiveness of the methods.

Journal ArticleDOI
TL;DR: In this article, the authors proposed a mesh-free method based on the moving least squares (MLS) meshless approach for numerical solution of two-dimensional (2D) variable-order time fractional nonlinear diffusion-wave equation (V-OTFND-WE) on arbitrary domains.

Journal ArticleDOI
TL;DR: In this paper, a numerical study was conducted using the finite difference technique to examine the mechanism of energy transfer as well as turbulence in the case of fully developed turbulent flow in a circular tube with constant wall temperature and heat flow conditions.
Abstract: A numerical study was conducted using the finite difference technique to examine the mechanism of energy transfer as well as turbulence in the case of fully developed turbulent flow in a circular tube with constant wall temperature and heat flow conditions. The methodology to solve this thermal problem is based on the energy equation, a fluid of constant properties in an axisymmetric and two-dimensional stationary flow. The global equations and the initial and boundary conditions acting on the problem are configured in dimensionless form in order to predict the characteristics of the turbulent fluid flow inside the tube. Using Thomas’ algorithm, a program in FORTRAN was developed to numerically solve the discretized form of the system of equations describing the problem. Finally, using this elaborate program, we were able to simulate the flow characteristics, for changing parameters such as Reynolds, Prandtl, and Peclet numbers along the pipe to obtain the important thermal model. These are discussed in detail in this work. Comparison of the results to published data shows that results are a good match to the published quantities.

Journal ArticleDOI
TL;DR: In this paper, a stress integration algorithm based on finite difference method (FDM) was proposed to effectively deal with both first and second derivatives of yield and potential functions which are the lengthiest component in stress integration procedure.

Journal ArticleDOI
TL;DR: It is proved that the convergence rate of constant-coefficient initial-boundary value problems, with a first or second derivative in time and a highest spatial derivative of order q, and their semi-discrete finite difference approximations is min ⁡.

Posted Content
TL;DR: This paper analyzes several methods for approximating gradients of noisy functions using only function values, including finite differences, linear interpolation, Gaussian smoothing, and smoothing on a sphere, and derives bounds on the number of samples and the sampling radius which guarantee favorable convergence properties for a line search or fixed step size descent method.
Abstract: In this paper, we analyze several methods for approximating gradients of noisy functions using only function values. These methods include finite differences, linear interpolation, Gaussian smoothing and smoothing on a sphere. The methods differ in the number of functions sampled, the choice of the sample points, and the way in which the gradient approximations are derived. For each method, we derive bounds on the number of samples and the sampling radius which guarantee favorable convergence properties for a line search or fixed step size descent method. To this end, we use the results in [Berahas et al., 2019] and show how each method can satisfy the sufficient conditions, possibly only with some sufficiently large probability at each iteration, as happens to be the case with Gaussian smoothing and smoothing on a sphere. Finally, we present numerical results evaluating the quality of the gradient approximations as well as their performance in conjunction with a line search derivative-free optimization algorithm.

Journal ArticleDOI
TL;DR: In this article, a truncated Taylor series expansion was applied to the Kirchhoff head at the material interface to avoid the dyadic characteristics at the interface between different soil types, which could be used to simulate complicated heterogeneous flow at a large-scale watershed or regional scale.

Journal ArticleDOI
TL;DR: A parameter-uniform numerical method is constructed and analysed for solving one-dimensional singularly perturbed parabolic problems with two small parameters and is proved to be uniformly convergent of order two in both the spatial and temporal variables.
Abstract: In the present paper, a parameter-uniform numerical method is constructed and analysed for solving one-dimensional singularly perturbed parabolic problems with two small parameters. The solution of this class of problems may exhibit exponential (or parabolic) boundary layers at both the left and right part of the lateral surface of the domain. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution, we consider the implicit Euler method for time stepping on a uniform mesh and a special hybrid monotone difference operator for spatial discretization on a specially designed piecewise uniform Shishkin mesh. The resulting scheme is shown to be first-order convergent in temporal direction and almost second-order convergent in spatial direction. We then improve the order of convergence in time by means of the Richardson extrapolation technique used in temporal variable only. The resulting scheme is...

Journal ArticleDOI
TL;DR: An explicit second order staggered finite difference discretization scheme for forward simulation of natural gas transport in pipeline networks and derives compatibility conditions for linking domain boundary values to enable efficient, explicit simulation of gas flows propagating through a network with pressure changes created by gas compressors.

Journal ArticleDOI
TL;DR: In this article, a numerical scheme is implemented to approximate the solution of time fractional stochastic Korteweg-de Vries (KdV) equation, which is based on a meshless method based on radial basis functions (RBFs).
Abstract: In this article, a numerical scheme is implemented to approximate the solution of time fractional stochastic Korteweg–de Vries (KdV) equation. The structure of the proposed method is such that it first employ finite difference technique to transform the time fractional stochastic KdV equation into elliptic stochastic differential equations (SDEs). Then resulting elliptic SDEs has been estimated via a meshless method based on radial basis functions (RBFs). It should be mentioned that every RBFs with sufficient smoothness can be applied. In this paper, we use Gaussian RBFs which are infinitely smoothness to approximate the functions in the resulting elliptic SDEs. The most important advantage of proposed method respect to traditional numerical method is that the noise terms are directly simulated at the collocation points in each step time. Finally, some test problems are presented to investigate the performance and accuracy of the new method.

Journal ArticleDOI
TL;DR: Moczo and de la Puente as mentioned in this paper proposed a method to solve the same problem in the context of the supercomputing center of the Barcelona Supercomputing Center, Nexus II-Planta 3C, Jordi Girona, Spain.
Abstract: Peter Moczo,1,2 David Gregor,1 Jozef Kristek 1,2 and Josep de la Puente3 1Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, Mlynska dolina F1, 84248 Bratislava, Slovak Republic. E-mail: moczo@fmph.uniba.sk 2Earth Science Institute, Slovak Academy of Sciences, Dubravska cesta 9, 84528 Bratislava, Slovak Republic 3Barcelona Supercomputing Center, Nexus II – Planta 3C, Jordi Girona 29, 08034 Barcelona, Spain

Journal ArticleDOI
TL;DR: The analytical study shows that the presented scheme is unconditionally stable and convergent and some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique.

Journal ArticleDOI
TL;DR: In this article, a novel unconditionally positivity preserving finite difference (FD) scheme was proposed for the solution of a three dimensional Brusselator reaction-diffusion system, and the Von Neumann stability method and Taylor series expansion was applied to verify unconditional stability and consistency of the proposed FD scheme.
Abstract: In many mathematical models, positivity is one of the attributes that must be possessed by the continuous systems. For instance, the unknown quantities in the Brusselator reaction-diffusion model represent the concentration of two reactant species. The negative values of concentration produced by any numerical methods is meaningless. This work is concerned with the investigation of a novel unconditionally positivity preserving finite difference (FD) scheme to be used for the solution of three dimensional Brusselator reaction-diffusion system. Von Neumann stability method and Taylor series expansion is applied to verify unconditional stability and consistency of the proposed FD scheme. Results are compared against well-known forward Euler FD scheme and some results reported in the literature.

Journal ArticleDOI
TL;DR: A novel technique for the imposition of non-linear entropy conservative and entropy stable solid wall boundary conditions for the compressible Navier-Stokes equations in the presence of an adiabatic wall, or a wall with a prescribed heat entropy flow is presented.