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1.
In this paper, a fast second‐order accurate difference scheme is proposed for solving the space–time fractional equation. The temporal Caputo derivative is approximated by ?L2 ‐1σ formula which employs the sum‐of‐exponential approximation to the kernel function appeared in Caputo derivative. The second‐order linear spline approximation is applied to the spatial Riemann–Liouville derivative. At each time step, a fast algorithm, the preconditioned conjugate gradient normal residual method with a circulant preconditioner (PCGNR), is used to solve the resulting system that reduces the storage and computational cost significantly. The unique solvability and unconditional convergence of the difference scheme are shown by the discrete energy method. Numerical examples are given to verify numerical accuracy and efficiency of the difference schemes. 相似文献
2.
Numerical simulation of a class of fractional subdiffusion equations via the alternating direction implicit method 下载免费PDF全文
Wenjuan Yao Jiebao Sun Boying Wu Shengzhu Shi 《Numerical Methods for Partial Differential Equations》2016,32(2):531-547
In this article, a new numerical technique is proposed for solving the two‐dimensional time fractional subdiffusion equation with nonhomogeneous terms. After a transformation of the original problem, standard central difference approximation is used for the spatial discretization. For the time step, a new fractional alternating direction implicit (FADI) scheme based on the L1 approximation is considered. This FADI scheme is constructed by adding a small term, so it is different from standard FADI methods. The solvability, unconditional stability and H1 norm convergence are proved. Finally, numerical examples show the effectiveness and accuracy of our proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 531–547, 2016 相似文献
3.
Qinxu Ding Patricia J. Y. Wong 《Numerical Methods for Partial Differential Equations》2021,37(1):643-673
In this paper, we derive a fourth order approximation for the generalized fractional derivative that is characterized by a scale function z(t) and a weight function w(t) . Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method, and it is shown that the temporal and spatial convergence orders are both 4. Several numerical experiments are provided to illustrate the efficiency of our scheme. 相似文献
4.
Lei Zhao Zhi‐zhong Sun Jian‐ming Liu 《Numerical Methods for Partial Differential Equations》2006,22(3):744-760
In this article, we present a numerical simulation of one‐dimensional problem of quasi‐static contact with an elastic obstacle. A finite difference scheme is derived by the method of reduction of order on uniform meshes. The stability and convergence are proved. The convergence order is of O(τ2 + h2), where τ and h are the time step size and the space step size, respectively. Some numerical examples demonstrate the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 相似文献
5.
Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa–Holm equation with different initial solitary waves 下载免费PDF全文
C. H. Yu Tony W. H. Sheu C. H. Chang S. J. Liao 《Numerical Methods for Partial Differential Equations》2015,31(5):1645-1664
In this article, the solution of Camassa–Holm (CH) equation is solved by the proposed two‐step method. In the first step, the sixth‐order spatially accurate upwinding combined compact difference scheme with minimized phase error is developed in a stencil of four points to approximate the first‐order derivative term. For the purpose of retaining both of the long‐term accurate Hamiltonian property and the geometric structure inherited in the CH equation, the time integrator used in this study should be able to conserve symplecticity. In the second step, the Helmholtz equation governing the pressure‐like variable is approximated by the sixth‐order accurate three‐point centered compact difference scheme. Through the fundamental and numerical verification studies, the integrity of the proposed high‐order scheme is demonstrated. Another aim of this study is to reveal the wave propagation nature for the investigated shallow water equation subject to different initial wave profiles, whose peaks take the smooth, peakon, and cuspon forms. The transport phenomena for the cases with/without inclusion of the linear first‐order advection term κux in the CH equation will be addressed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1645–1664, 2015 相似文献
6.
G. I. Shishkin L. P. Shishkina 《Computational Mathematics and Mathematical Physics》2010,50(3):437-456
The Dirichlet problem on a vertical strip is examined for a singularly perturbed semilinear elliptic convection-diffusion
equation. For this problem, the basic nonlinear difference scheme based on the classical approximations on piecewise uniform
grids condensing in the vicinity of boundary layers converges ɛ-uniformly with an order at most almost one. The Richardson
technique is used to construct a nonlinear scheme that converges ɛ-uniformly with an improved order, namely, at the rate O(N
1−2ln2
N
1 + N
2−2), where N
1 + 1 and N
2 + 1 are the number of grid nodes along the x
1-axis and per unit interval of the x
2-axis, respectively. This nonlinear basic scheme underlies the linearized iterative scheme, in which the nonlinear term is
calculated using the values of the sought function found at the preceding iteration step. The latter scheme is used to construct
a linearized iterative Richardson scheme converging ɛ-uniformly with an improved order. Both the basic and improved iterative
schemes converge ɛ-uniformly at the rate of a geometric progression as the number of iteration steps grows. The upper and
lower solutions to the iterative Richardson schemes are used as indicators, which makes it possible to determine the iteration
step at which the same ɛ-uniform accuracy is attained as that of the non-iterative nonlinear Richardson scheme. It is shown
that no Richardson schemes exist for the convection-diffusion boundary value problem converging ɛ-uniformly with an order
greater than two. Principles are discussed on which the construction of schemes of order greater than two can be based. 相似文献
7.
Jeff Borggaard Traian Iliescu John Paul Roop 《Numerical Methods for Partial Differential Equations》2012,28(3):1056-1078
The r‐Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two‐level finite element approximation schemes applied to the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity, where r is the order of the power‐law artificial viscosity term. In the two‐level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single‐step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two‐level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide rigorous numerical analysis of the two‐level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H, fine mesh size h, and filter radius δ. We also investigate the two‐level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward‐facing step at Reynolds number Re = 5100. In all numerical tests, the two‐level algorithm was proven to achieve the same order of accuracy as the standard one‐level algorithm, at a fraction of the computational cost. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011 相似文献
8.
Guang‐Hua Gao Zhi‐Zhong Sun 《Numerical Methods for Partial Differential Equations》2013,29(5):1459-1486
This is the further work on compact finite difference schemes for heat equation with Neumann boundary conditions subsequent to the paper, [Sun, Numer Methods Partial Differential Equations (NMPDE) 25 (2009), 1320–1341]. A different compact difference scheme for the one‐dimensional linear heat equation is developed. Truncation errors of the proposed scheme are O(τ2 + h4) for interior mesh point approximation and O(τ2 + h3) for the boundary condition approximation with the uniform partition. The new obtained scheme is similar to the one given by Liao et al. (NMPDE 22 (2006), 600–616), while the major difference lies in no extension of source terms to outside the computational domain any longer. Compared with ones obtained by Zhao et al. (NMPDE 23 (2007), 949–959) and Dai (NMPDE 27 (2011), 436–446), numerical solutions at all mesh points including two boundary points are computed in our new scheme. The significant advantage of this work is to provide a rigorous analysis of convergence order for the obtained compact difference scheme using discrete energy method. The global accuracy is O(τ2 + h4) in discrete maximum norm, although the spatial approximation order at the Neumann boundary is one lower than that for interior mesh points. The analytical techniques are important and can be successfully used to solve the open problem presented by Sun (NMPDE 25 (2009), 1320–1341), where analyzed theoretical convergence order of the scheme by Liao et al. (NMPDE 22 (2006), 600–616) is only O(τ2 + h3.5) while the numerical accuracy is O(τ2 + h4), and convergence order of theoretical analysis for the scheme by Zhao et al. (NMPDE 23 (2007), 949–959) is O(τ2 + h2.5), while the actual numerical accuracy is O(τ2 + h3). Following the procedure used for the new obtained difference scheme in this work, convergence orders of these two schemes can be proved rigorously to be O(τ2 + h4) and O(τ2 + h3), respectively. Meanwhile, extension to the case involving the nonlinear reaction term is also discussed, and the global convergence order O(τ2 + h4) is proved. A compact ADI difference scheme for solving two‐dimensional case is derived. Finally, several examples are given to demonstrate the numerical accuracy of new obtained compact difference schemes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
9.
In this paper, we study a numerical scheme to solve coupled Maxwell's equations with a nonlinear conductivity. This model plays an important role in the study of type‐II superconductors. The approximation scheme is based on backward Euler discretization in time and mixed conforming finite elements in space. We will prove convergence of this scheme to the unique weak solution of the problem and develop the corresponding error estimates. As a next step, we study the stability of the scheme in the quasi‐static limit ? → 0 and present the corresponding convergence rate. Finally, we support the theory by several numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
10.
Rumeng Zheng Xiaoyun Jiang Hui Zhang 《Mathematical Methods in the Applied Sciences》2020,43(5):2216-2232
In the current paper, a heat transfer model is suggested based on a time-fractional dual-phase-lag (DPL) model. We discuss the model in two parts, the direct problem and the inverse problem. Firstly, for solving it, the finite difference/Legendre spectral method is constructed. In the temporal direction, we employ the weighted and shifted Grünwald approximation, which can achieve second order convergence. In the spatial direction, the Legendre spectral method is used, it can obtain spectral accuracy. The stability and convergence are theoretically analyzed. For the inverse problem, the Bayesian method is used to construct an algorithm to estimate the four parameters for the model, namely, the time-fractional order α, the time-fractional order β, the delay time τT, and the relaxation time τq. Next, numerical experiments are provided to demonstrate the effectiveness of our scheme, with the values of τq and τT for processed meat employed. We also make a comparison with another method. The data obtained for the direct problem are used in the parameter estimation. The paper provides an accurate and efficient numerical method for the time-fractional DPL model. 相似文献
11.
Hai‐Yan Cao Zhi‐Zhong Sun Guang‐Hua Gao 《Numerical Methods for Partial Differential Equations》2014,30(2):451-471
The Camassa–Holm (CH) system is a strong nonlinear third‐order evolution equation. So far, the numerical methods for solving this problem are only a few. This article deals with the finite difference solution to the CH equation. A three‐level linearized finite difference scheme is derived. The scheme is proved to be conservative, uniquely solvable, and conditionally second‐order convergent in both time and space in the discrete L∞ norm. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 451–471, 2014 相似文献
12.
Werner Varnhorn 《Mathematical Methods in the Applied Sciences》1992,15(2):89-108
In the present paper we use a time delay ? > 0 for an energy conserving approximation of the non-linear term of the non-stationary Navier–Stokes equations. We prove that the corresponding initial-value problem (N?) in smoothly bounded domains G ? ?3 is well-posed. We study a semidiscretized difference scheme for (N?) and prove convergence to optimal order in the Sobolev space H2(G). Passing to the limit ?→0 we show that the sequence of stabilized solutions has an accumulation point such that it solves the Navier–Stokes problem (No) in a weak sense (Hopf). 相似文献
13.
《Mathematical Methods in the Applied Sciences》2018,41(5):2119-2139
In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P1−P1 or P1−P0) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results. 相似文献
14.
In this paper, we consider a space-time Riesz–Caputo fractional advection-diffusion equation. The equation is obtained from
the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative
of order α ∈ (0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β
1 ∈ (0,1) and β
2 ∈ (1,2], respectively. We present an explicit difference approximation and an implicit difference approximation for the equation
with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit difference
approximation is unconditionally stable and convergent, but the explicit difference approximation is conditionally stable
and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order
accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example
is given; the numerical results are in good agreement with theoretical analysis. 相似文献
15.
Ling‐yun Zhang Zhi‐zhong Sun 《Numerical Methods for Partial Differential Equations》2004,20(2):230-247
A linearized three‐level difference scheme on nonuniform meshes is derived by the method of the reduction of order for the Neumann boundary value problem of a nonlinear parabolic system. It is proved that the difference scheme is uniquely solvable and second‐order convergent in L∞‐norm. A numerical example is given. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 230–247, 2004 相似文献
16.
This paper is concerned with the numerical solution of the Cauchy problem for the Benjamin-Ono equationu
t
+uu
x
−Hu
xx
=0, whereH denotes the Hilbert transform. Our numerical method first approximates this Cauchy problem by an initial-value problem for
a corresponding 2L-periodic problem in the spatial variable, withL large. This periodic problem is then solved using the Crank-Nicolson approximation in time and finite difference approximations
in space, treating the nonlinear term in a standard conservative fashion, and the Hilbert transform by a quadrature formula
which may be computed efficiently using the Fast Fourier Transform. 相似文献
17.
A fully discrete spectral method for fractional Cattaneo equation based on Caputo–Fabrizo derivative
Haonan Li Shujuan Lü Tao Xu 《Numerical Methods for Partial Differential Equations》2019,35(3):936-954
Recently Caputo and Fabrizio introduced a new derivative with fractional order without singular kernel. The derivative can be used to describe the material heterogeneities and the fluctuations of different scales. In this article, we derived a new discretization of Caputo–Fabrizio derivative of order α (1 < α < 2) and applied it into the Cattaneo equation. A fully discrete scheme based on finite difference method in time and Legendre spectral approximation in space is proposed. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in H1 norm is O(τ2 + N1?m), where τ, N and m are the time‐step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Furthermore, the accuracy and applicability of the scheme are confirmed by numerical examples to support the theoretical results. 相似文献
18.
V. N. Mel'nik 《Journal of Mathematical Sciences》1993,66(6):2642-2646
For numerical solution of the coupled one-dimensional problem of dynamic thermoelasticity in stresses (strains) we construct a second-order approximating difference scheme. We study its stability and obtain an a priori estimate. We prove that the solution of the scheme converges to a generalized solution of the original problem in the Sobolev class W
2
2
(QT).Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 95–99. 相似文献
19.
In this study, we develop a fourth‐order compact finite difference scheme for solving a model of energy exchanges in a generalized N‐carrier system with heat sources and Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for microheat transfer. By using the matrix analysis, the compact finite difference numerical scheme is shown to be unconditionally stable. The accuracy of the solution obtained by the scheme is tested by a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 相似文献
20.
In this paper, we consider the numerical approximation for the fractional diffusion-wave equation. The main purpose of this paper is to solve and analyze this problem by introducing an implicit fully discrete local discontinuous Galerkin method. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. By choosing the numerical fluxes carefully we prove that our scheme is unconditionally stable and get L 2 error estimates of \(O(h^{k+1}+(\Delta t)^{2}+(\Delta t)^{\frac {\alpha }{2}}h^{k+1})\) . Finally numerical examples are performed to illustrate the efficiency and the accuracy of the method. 相似文献