Showing papers on "Finite difference published in 2022"
TL;DR: The generalized finite difference method (GFDM) as discussed by the authors is a typical meshless collocation method based on the Taylor series expansion and the moving least squares technique, and the theoretical results of the meshless function approximation in the GFDM are studied theoretically, and a stabilized approximation is proposed by revising the computational formulas of the original approximation.
Abstract: The generalized finite difference method (GFDM) is a typical meshless collocation method based on the Taylor series expansion and the moving least squares technique. In this paper, we first provide theoretical results of the meshless function approximation in the GFDM. Properties, stability and error estimation of the approximation are studied theoretically, and a stabilized approximation is proposed by revising the computational formulas of the original approximation. Then, we provide theoretical results consisting of error bound and condition number of the GFDM. Numerical results are finally provided to confirm these theoretical results.
28 citations
TL;DR: A novel meshless technique for solving a class of fractional differential equations based on moving least squares and on the existence of a fractional Taylor series for Caputo derivatives is presented.
25 citations
TL;DR: A strong-form local meshless approach established on radial basis function-finite difference (RBF-FD) method for spatial approximation is developed in this article, where polyharmonic splines are used as radial basis functions with augmented polynomials.
21 citations
TL;DR: In this paper, the numerical solution of a time fractional partial integro-differential equation of Volterra type, where the time derivative is defined in Caputo sense, is studied.
16 citations
TL;DR: In this paper, a novel application of integrated evolutionary computing paradigm is presented for the analysis of nonlinear systems of differential equations representing the dynamics of virus propagation model in computer networks by exploiting the discretization strength of finite difference procedure, global search efficacy of GA aided with interior-point method (IPM) as efficient local search mechanism.
16 citations
TL;DR: In this paper, a class of high-order finite difference schemes with minimized dispersion and adaptive dissipation is proposed, where a scale sensor is devised to quantify the local length scale of the numerical solution as the effective scaled wavenumber.
14 citations
TL;DR: In this paper , it was shown that a lattice Boltzmann scheme can be rewritten as a multi-step Finite Difference scheme on the conserved variables, and that the notion of consistency of the corresponding finite difference scheme allows to invoke the Lax-Richtmyer theorem in the case of linear lattice schemes.
Abstract: Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of consistency with respect to the target equations and to provide a precise notion of stability. In order to alleviate these shortages and introduce a rigorous framework, we demonstrate that any lattice Boltzmann scheme can be rewritten as a corresponding multi-step Finite Difference scheme on the conserved variables. This is achieved by devising a suitable formalism based on operators, commutative algebra and polynomials. Therefore, the notion of consistency of the corresponding Finite Difference scheme allows to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequently-used von Neumann-like stability analysis for lattice Boltzmann schemes entirely corresponds to the von Neumann stability analysis of their Finite Difference counterpart. More generally, the usual tools for the analysis of Finite Difference schemes are now readily available to study lattice Boltzmann schemes. Their relevance is verified by means of numerical illustrations.
11 citations
TL;DR: In this article , the authors compared the results of the finite volume method and the finite difference method using the Burgers equation and the Buckley-Leverett equation as examples to simulate the previously mentioned methods.
Abstract: In this paper, we present an intensive investigation of the finite volume method (FVM) compared to the finite difference methods (FDMs). In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov’s scheme to simulate the results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods.
11 citations
TL;DR: In this article , the importance of high-frequency damping in high-order conservative finite-difference schemes for viscous terms in the Navier-Stokes equations is discussed, and a modification to the viscous scheme rather than the inviscid scheme resolves a problem with spurious oscillations around shocks.
11 citations
10 citations
TL;DR: In this paper , the authors developed the enhanced unconditionally positive finite difference method (EUPFD), and use it to solve linear and nonlinear advection-diffusion-reaction (ADR) equations.
Abstract: In this study, we develop the enhanced unconditionally positive finite difference method (EUPFD), and use it to solve linear and nonlinear advection–diffusion–reaction (ADR) equations. This method incorporates the proper orthogonal decomposition technique to the unconditionally positive finite difference method (UPFD) to reduce the degree of freedom of the ADR equations. We investigate the efficiency and effectiveness of the proposed method by checking the error, convergence rate, and computational time that the method takes to converge to the exact solution. Solutions obtained by the EUPFD were compared with the exact solutions for validation purposes. The agreement between the solutions means the proposed method effectively solved the ADR equations. The numerical results show that the proposed method greatly improves computational efficiency without a significant loss in accuracy for solving linear and nonlinear ADR equations.
TL;DR: In this paper, the authors developed a new efficient and accurate 2D numerical model for the transport of dense contaminants in unsaturated porous media that allows for the simulation of large-scale problems.
TL;DR: In this paper , the authors employed the finite element method for heat and mass transfer of MHD Maxwell nanofluid flow over the stretching sheet under the effects of radiations and chemical reactions.
Abstract: The recent study was concerned with employing the finite element method for heat and mass transfer of MHD Maxwell nanofluid flow over the stretching sheet under the effects of radiations and chemical reactions. Moreover, the effects of viscous dissipation and porous plate were considered. The mathematical model of the flow was described in the form of a set of partial differential equations (PDEs). Further, these PDEs were transformed into a set of nonlinear ordinary differential equations (ODEs) using similarity transformations. Rather than analytical integrations, numerical integration was used to compute integrals obtained by applying the finite element method. The mesh-free analysis and comparison of the finite element method with the finite difference method are also provided to justify the calculated results. The effect of different parameters on velocity, temperature and concentration profile is shown in graphs, and numerical values for physical quantities of interest are also given in a tabular form. In addition, simulations were carried out by employing software that applies the finite element method for solving PDEs. The calculated results are also portrayed in graphs with varying sheet velocities. The results show that the second-order finite difference method is more accurate than the finite element method with linear interpolation polynomial. However, the finite element method requires less number of iterations than the finite difference method in a considered particular case. We had high hopes that this work would act as a roadmap for future researchers entrusted with resolving outstanding challenges in the realm of enclosures utilized in industry and engineering.
TL;DR: In this article , the authors use known methods in both real and Fourier space to derive finite-difference approximations of leading differential terms in 2D and 3D that are isotropic at order $h^2$ of the lattice spacing.
TL;DR: In this paper , high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography.
TL;DR: In this paper , the authors considered a singularly perturbed reaction-diffusion equation with a discontinuous source term and constructed a numerical approach adopting an efficient hybrid finite difference method that includes a proper layer adapted piece-wise uniform Shishkin mesh.
Abstract: This paper considers a two-dimensional singularly perturbed reaction-diffusion equation with a discontinuous source term. Due to this discontinuity, interior, corner, and boundary layers appear in the solution for adequately small values of the perturbation parameter ϵ . To achieve a decent estimate of the solution, we construct a numerical approach adopting an efficient hybrid finite difference method that includes a proper layer adapted piece-wise uniform Shishkin mesh. Further, we prove that the hybrid finite difference method is almost second-order uniformly convergent with respect to the perturbation parameter. We have implemented our method to test examples. Numerical results are verifying the theoretical results.
TL;DR: In this paper, the weak solvability of a macroscopic quasilinear reaction-diffusion system posed in a 2D porous medium which undergoes microstructural problems is studied.
Abstract: We study the weak solvability of a macroscopic, quasilinear reaction–diffusion system posed in a 2 D porous medium which undergoes microstructural problems. The solid matrix of this porous medium is assumed to be made out of circles of not-necessarily uniform radius. The growth or shrinkage of these circles, which are governed by an ODE, has direct feedback to the macroscopic diffusivity via an additional elliptic cell problem. The reaction–diffusion system describes the macroscopic diffusion, aggregation, and deposition of populations of colloidal particles of various sizes inside a porous media made of prescribed arrangement of balls. The mathematical analysis of this two-scale problem relies on a suitable application of Schauder’s fixed point theorem which also provides a convergent algorithm for an iteration method to compute finite difference approximations of smooth solutions to our multiscale model. Numerical simulations illustrate the behavior of the local concentration of the colloidal populations close to clogging situations.
TL;DR: In this article , a space-time generalized finite difference scheme is proposed to effectively solve the unsteady double-diffusive natural convection problem in the fluid-saturated porous media.
Abstract: In this paper, the space-time generalized finite difference scheme is proposed to effectively solve the unsteady double-diffusive natural convection problem in the fluid-saturated porous media. In such a case, it is mathematically described by nonlinear time-dependent partial differential equations based on Darcy's law. In this work, the space-time approach is applied using a combination of the generalized finite difference, Newton-Raphson, and time-marching methods. In the space-time generalized finite difference scheme, the spatial and temporal derivatives can be performed using the technique for spatial discretization. Thus, the stability of the proposed numerical scheme is determined by the generalized finite difference method. Due to the property of this numerical method, which is based on the Taylor series expansion and the moving-least square method, the resultant matrix system is a sparse matrix. Then, the Newton-Raphson method is used to solve the nonlinear system efficiently. Furthermore, the time-marching method is utilized to proceed along the time axis after a numerical process in one space-time domain. By using this method, the proposed numerical scheme can efficiently simulate the problems which have an unpredictable end time. In this study, three benchmark examples are tested to verify the capability of the proposed meshless scheme.
TL;DR: In this article, the authors applied an unconditionally stable half-sweep finite difference approach to solve the time-fractional diffusion equation in a one-dimensional model, where a Caputo fractional operator was used to substitute the time fractional derivative term approximately.
Abstract: Solving time-fractional diffusion equation using a numerical method
has become a research trend nowadays since analytical approaches are quite limited. There is increasing usage of the finite difference method, but the efficiency of
the scheme still needs to be explored. A half-sweep finite difference scheme is
well-known as a computational complexity reduction approach. Therefore, the
present paper applied an unconditionally stable half-sweep finite difference
scheme to solve the time-fractional diffusion equation in a one-dimensional model. Throughout this paper, a Caputo fractional operator is used to substitute the
time-fractional derivative term approximately. Then, the stability of the difference
scheme combining the half-sweep finite difference for spatial discretization and
Caputo time-fractional derivative is analyzed for its compatibility. From the formulated half-sweep Caputo approximation to the time-fractional diffusion equation, a linear system corresponds to the equation contains a large and sparse
coefficient matrix that needs to be solved efficiently. We construct a preconditioned matrix based on the first matrix and develop a preconditioned accelerated
over relaxation (PAOR) algorithm to achieve a high convergence solution. The
convergence of the developed method is analyzed. Finally, some numerical
experiments from our research are given to illustrate the efficiency of our computational approach to solve the proposed problems of time-fractional diffusion. The
combination of a half-sweep finite difference scheme and PAOR algorithm can be
a good alternative computational approach to solve the time-fractional diffusion
equation-based mathematical physics model.
TL;DR: In this article , the authors compare the long-time error bounds and spatial resolution of finite difference methods with different spatial discretizations for the Dirac equation with small electromagnetic potentials characterized by ǫ∈(0, 1] a dimensionless parameter.
TL;DR: In this paper , a bisection search for finding a finite-difference interval that balances the truncation error, which arises from the error in the Taylor series approximation, and the measurement error which results from noise in the function evaluation is proposed.
Abstract: A common approach for minimizing a smooth nonlinear function is to employ finite-difference approximations to the gradient. While this can be easily performed when no error is present within the function evaluations, when the function is noisy, the optimal choice requires information about the noise level and higher-order derivatives of the function, which is often unavailable. Given the noise level of the function, we propose a bisection search for finding a finite-difference interval for any finite-difference scheme that balances the truncation error, which arises from the error in the Taylor series approximation, and the measurement error, which results from noise in the function evaluation. Our procedure produces reliable estimates of the finite-difference interval at low cost without explicitly approximating higher-order derivatives. We show its numerical reliability and accuracy on a set of test problems. When combined with limited memory BFGS, we obtain a robust method for minimizing noisy black-box functions, as illustrated on a subset of unconstrained CUTEst problems with synthetically added noise.
TL;DR: In this article , the authors presented a novel approach for seismic modeling combining conventional finite differences with deep neural networks, which includes the following steps: First, a training dataset composed of a small number of common-shot gathers is generated.
Abstract: In this study, we present a novel approach for seismic modeling combining conventional finite differences with deep neural networks. The method includes the following steps: First, a training dataset composed of a small number of common-shot gathers is generated. The dataset is computed using a finite-difference scheme with fine spatial and temporal discretization. Second, the entire set of common-shot seismograms is generated using an inaccurate numerical method, such as a finite difference scheme on a coarse mesh. Third, the numerical dispersion mitigation neural network is trained and applied to the entire dataset to suppress the numerical dispersion. We tested the approach on two 2D models, illustrating a significant acceleration of seismic modeling.
TL;DR: In this paper , the authors studied the heroin epidemic model's dynamics based on the law of mass action and presented the essential features of the model like positivity, boundedness, equilibria, and reproduction number.
Abstract: Numerical modelling of real-world problems has a significant role in different disciplines like biology, physics, chemistry, economics, and many more. The numerical results are the authentication of the results of the dynamical analysis. We studied the heroin epidemic model's dynamics based on the law of mass action. The essential features of the model like positivity, boundedness, equilibria, and reproduction number are presented. Some standard and nonstandard computational methods like Euler, Runge-Kutta, and Nonstandard finite difference (NSFD) are given for the said model. The nonstandard finite difference method was developed to restore the fundamental properties of biological problems. Also, the proposed numerical model is consistent with the behaviour of the continuous model. To that end, simulations for the comparison of computational methods are presented.
TL;DR: In this article, the plane-wave destruction (PWD) method is used to estimate the local slope of seismic images, which has a variety of meaningful applications and is a widely accepted technique in t...
Abstract: The local slope estimated from seismic images has a variety of meaningful applications. Slope estimation based on the plane-wave destruction (PWD) method is a widely accepted technique in t...
01 Jan 2022
TL;DR: In this article , the authors applied an unconditionally stable half-sweep finite difference scheme to solve the time-fractional diffusion equation in a one-dimensional model, where a Caputo fractional operator was used to substitute the time fractional derivative term approximately.
Abstract: Solving time-fractional diffusion equation using a numerical method has become a research trend nowadays since analytical approaches are quite limited. There is increasing usage of the finite difference method, but the efficiency of the scheme still needs to be explored. A half-sweep finite difference scheme is well-known as a computational complexity reduction approach. Therefore, the present paper applied an unconditionally stable half-sweep finite difference scheme to solve the time-fractional diffusion equation in a one-dimensional model. Throughout this paper, a Caputo fractional operator is used to substitute the time-fractional derivative term approximately. Then, the stability of the difference scheme combining the half-sweep finite difference for spatial discretization and Caputo time-fractional derivative is analyzed for its compatibility. From the formulated half-sweep Caputo approximation to the time-fractional diffusion equation, a linear system corresponds to the equation contains a large and sparse coefficient matrix that needs to be solved efficiently. We construct a preconditioned matrix based on the first matrix and develop a preconditioned accelerated over relaxation (PAOR) algorithm to achieve a high convergence solution. The convergence of the developed method is analyzed. Finally, some numerical experiments from our research are given to illustrate the efficiency of our computational approach to solve the proposed problems of time-fractional diffusion. The combination of a half-sweep finite difference scheme and PAOR algorithm can be a good alternative computational approach to solve the time-fractional diffusion equation-based mathematical physics model.
TL;DR: In this article , a high-order compact finite difference method is used to solve boundary value problems with Robin boundary conditions, which produces highly accurate results, which are displayed in tables and graphs.
Abstract: In this study, a high-order compact finite difference method is used to solve boundary value problems with Robin boundary conditions. The norm is to use a first-order finite difference scheme to approximate Neumann and Robin boundary conditions, but that compromises the accuracy of the entire scheme. As a result, new higher-order finite difference schemes for approximating Robin boundary conditions are developed in this work. Six examples for testing the applicability and performance of the method are considered. Convergence analysis is provided, and it is consistent with the numerical results. The results are compared with the exact solutions and published results from other methods. The method produces highly accurate results, which are displayed in tables and graphs.
TL;DR: In this article , an improved integrated radial basis function method based on the finite difference technique to approximate the second-order mixed partial derivatives with respect to x and y to get more accurate numerical results is presented.
Abstract: We present a method based on integrated radial basis function-finite difference for numerical solution of plane elastostatic equations which is a boundary value problem. The two-dimensional version of the governed equation is solved by the proposed method on various geometries such as the rectangular and irregular domains. In the current paper, one of our goals is to present an improved integrated radial basis function method based on the finite difference technique to approximate the second-order mixed partial derivatives with respect to x and y to get more accurate numerical results. Several examples are solved by applying integrated radial basis function based on finite difference method to check its accuracy and validity.
TL;DR: In this article, an improved integrated radial basis function method based on the finite difference technique to approximate the second-order mixed partial derivatives with respect to x and y to get more accurate numerical results is presented.
Abstract: We present a method based on integrated radial basis function-finite difference for numerical solution of plane elastostatic equations which is a boundary value problem. The two-dimensional version of the governed equation is solved by the proposed method on various geometries such as the rectangular and irregular domains. In the current paper, one of our goals is to present an improved integrated radial basis function method based on the finite difference technique to approximate the second-order mixed partial derivatives with respect to x and y to get more accurate numerical results. Several examples are solved by applying integrated radial basis function based on finite difference method to check its accuracy and validity.
01 Jan 2022
TL;DR: In this article, a Multilayer (ML) method for solving one-factor parabolic equations is proposed. But the method is not suitable for solving the multilayer heat equations, which are known in physics for a relatively long time but never used when solving financial problems.
Abstract: In this paper, we develop a Multilayer (ML) method for solving one-factor parabolic equations. Our approach provides a powerful alternative to the well-known finite difference and Monte Carlo methods. We discuss various advantages of this approach, which judiciously combines semi-analytical and numerical techniques and provides a fast and accurate way of finding solutions to the corresponding equations. To introduce the core of the method, we consider multilayer heat equations, known in physics for a relatively long time but never used when solving financial problems. Thus, we expand the analytic machinery of quantitative finance by augmenting it with the ML method. We demonstrate how one can solve various problems of mathematical finance by using our approach. Specifically, we develop efficient algorithms for pricing barrier options for time-dependent one-factor short-rate models, such as Black-Karasinski and Verhulst. Besides, we show how to solve the well-known Dupire equation quickly and accurately. Numerical examples confirm that our approach is considerably more efficient for solving the corresponding partial differential equations than the conventional finite difference method by being much faster and more accurate than the known alternatives.
TL;DR: In this paper, a nonstandard finite difference (NSFD) scheme for the improved hepatitis B virus (HBV) model is proposed and analyzed, which preserves positivity, boundedness and asymptotic stability of the model for all finite step sizes.