Convergence Of Difference Scheme Research Articles

Unconditionally Stable and Convergent Difference Scheme for Superdiffusion with Extrapolation

Approximating the Hadamard finite-part integral by the quadratic interpolation polynomials, we obtain a scheme for approximating the Riemann-Liouville fractional derivative of order α∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1, 2)$$\\end{document} and the error is shown to have the asymptotic expansion (d3τ3-α+d4τ4-α+d5τ5-α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\big ( d_{3} \ au ^{3- \\alpha } + d_{4} \ au ^{4-\\alpha } + d_{5} \ au ^{5-\\alpha } + \\cdots \\big ) + \\big ( d_{2}^{*} \ au ^{4} + d_{3}^{*} \ au ^{6} + d_{4}^{*} \ au ^{8} + \\cdots \\big ) $$\\end{document} at any fixed time, where τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} denotes the step size and dl,l=3,4,⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{l}, l=3, 4, \\dots $$\\end{document} and dl∗,l=2,3,⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{l}^{*}, l\\,=\\,2, 3, \\dots $$\\end{document} are some suitable constants. Applying the proposed scheme in temporal direction and the central difference scheme in spatial direction, a new finite difference method is developed for approximating the time fractional wave equation. The proposed method is unconditionally stable, convergent with order O(τ3-α),α∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O (\ au ^{3- \\alpha }), \\alpha \\in (1, 2)$$\\end{document} and the error has the asymptotic expansion. Richardson extrapolation is applied to improve the accuracy of the numerical method. The convergence orders are O(τ4-α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O ( \ au ^{4- \\alpha })$$\\end{document} and O(τ2(3-α)),α∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O ( \ au ^{2(3- \\alpha )}), \\alpha \\in (1, 2)$$\\end{document}, respectively, after first two extrapolations. Numerical examples are presented to show that the numerical results are consistent with the theoretical findings.

A novel distributed order time fractional model for heat conduction, anomalous diffusion, and viscoelastic flow problems

A novel distributed order time fractional model is constructed to solve heat conduction, anomalous diffusion and viscoelastic flow problems. Solutions of the formulated governing equation are obtained by the numerical discretization method that the mid-point quadrature rule is applied to approach the distributed order integrals and the L1-formula and L2-formula are used to discrete the time fractional derivatives. The stability and convergence of difference scheme with zero boundary conditions are studied and proven. In the end, three numerical examples are given. By employing an optimization algorithm, we obtain a good fitting result between the computational data with our model and heat conduction experimental data, which validates the effectiveness of the proposed model. Based on the background of the anomalous diffusion of volatile organic compound (VOC) adsorbed on building materials, the second example analyses the error and the convergence order by comparing the discrete solution with the exact solution, which validates the accuracy of the numerical difference method. Based on the solutions by using the difference method, the last example studies the unidirectional flows of generalized Oldroyd-B fluid driven by the moving plate or the pressure gradient and the effects of involved parameters on the velocity distribution are discussed graphically.

On convergence of difference schemes of high accuracy for one pseudo-parabolic Sobolev type equation

Difference schemes of the finite difference method and the finite element method of high-order accuracy in time and space are proposed and investigated for a pseudo-parabolic Sobolev type equation. The order of accuracy in space is improved in two ways using the finite difference method and the finite element method. The order of accuracy of the scheme in time is improved by a special discretization of the time variable. The corresponding a priori estimates are determined and, on their basis, the accuracy estimates of the proposed difference schemes are obtained with sufficient smoothness of the solution to the original differential problem. Algorithms for the implementation of the constructed difference schemes are proposed.

On the convergence of difference schemes of high accuracy for the equation of ion-acoustic waves in a magnetized plasma

Multiparametric difference schemes of the finite element method of a high order of accuracy for the Sobolevtype equation of the fourth-order in time are studied. In particular, the first boundary value problem for the equation of ion-acoustic waves in a magnetized plasma is considered. A high-order accuracy of the scheme is achieved due to the special discretization of time and space variables. The presence of parameters in the scheme makes it possible to regularize the accuracy of the schemes and optimize the implementation algorithm. An a priori estimate in a weak norm is obtained by the method of energy inequality. Based on this estimate and the Bramble-Hilbert lemma, the convergence of the constructed algorithms in classes of generalized solutions is proved. An algorithm for implementing the difference scheme is proposed.

Numerical Solution of the Two-Dimensional Elasticity Problem in Strains

Usually, the boundary value problems of the theory of elasticity are formulated with respect to displacements, and are reduced to the well-known Lame equations. Strains and stresses can be calculated from displacements as a solution to Lame's equation. Also known are the Beltrami Mitchell equations, which make it possible to formulate the boundary value problem of the theory of elasticity with respect to stresses. Currently, the boundary value problems of the theory of elasticity in stresses are studied in more detail in the two-dimensional case, and usually solved numerically with the introduction of the Airy stress function. But, the direct solution of boundary value problems of elasticity theory with respect to stresses requires further researches. This work, similarly to the boundary value problem in stresses, is devoted to the formulation and numerical solution of boundary value problems of the theory of elasticity with respect to deformations. The proposed boundary value problem consists of six Beltrami-Mitchell-type equations depending on strains and three equations of the equilibrium equation expressed with respect to deformations. As boundary conditions, in addition to the usual conditions for surface forces, three additional conditions are also introduced based on the equilibrium equations. The boundary value problem is considered in detail for a rectangular area. The discrete analogue of the boundary value problem is composed by the finite difference method. The convergence of difference schemes and an iterative method for their solution are studied. Software has been developed in the C++ environment for solving boundary value problems in the theory of elasticity and deformation. A number of boundary value problems on the deformation of a rectangular plate are solved numerically under various boundary conditions. The reliability of the obtained results is substantiated by comparing the numerical results, with the exact solution, as well as with the known solutions of the plate tension problems with parabolic and uniformly distributed edge loads.

High order compact difference scheme for solving the time multi-term fractional sub-diffusion equations

<abstract><p>In this paper, a high order compact finite difference is established for the time multi-term fractional sub-diffusion equation. The derived numerical differential formula can achieve second order accuracy in time and four order accuracy in space. A unconditionally stable and convergent difference scheme is presented, and a rigorous proof for the stability and convergence is given. Numerical results demonstrate the efficiency of the proposed difference schemes.</p></abstract>

Stability and Convergence of Difference Schemes Approximating Boundary Value Problems for Loaded Sobolev-Type Fractional Differential Equations

Stability and Convergence of Difference Schemes Approximating Boundary Value Problems for Loaded Sobolev-Type Fractional Differential Equations

Numerical analysis of the stability of nonequilibrium 3D evolutionary transfer processes in the network

Analytical methods for solving various problems of an applied nature (for example, non-stationary transfer problems over network hydro, gas and heat carriers), whose mathematical models use the formalisms of evolutionary differential systems, are possible with rare exceptions. That is why the construction of numerical and simulation models for the use of quantitative analysis methods becomes a universal research tool, if at the same time the implementation of these models on a computer is carried out – in other words, a complex of software engineering of the process under study is formed. The study uses the method of semidiscretization by a time variable of the mathematical model of the evolutionary non-equilibrium process of continuous medium transfer, which remains one of the most effective methods for analyzing applied problems. In this case, the elliptic operator of the mathematical model has a special basis (a system of eigenfunctions), which is why the analysis is reduced to the study of a boundary value problem for elliptic-type equations with a spatial variable changing on a network-like domain. The paper presents the conditions for unambiguous weak solvability of a differential-difference system, which is a difference analogue in the time variable of the original system, and the way of constructing an algorithm for finding an approximate solution is indicated. The study contains an analysis of the stability and convergence of difference schemes of evolutionary network-like nonequilibrium processes of continuous media transfer over network carriers and includes an analysis of the correctness of the mathematical model of this process. The results of the work are applicable in the framework of oil and gas engineering to the study of issues of stabilization and parametric optimization of the processes of transportation of liquid media through spatial networks.

Stable and convergent difference schemes for weakly singular convolution integrals

We obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order integrals using Taylor and fractional Taylor series expansions and grouping terms in a novel manner. A fractional Taylor series expansion argument is utilized to provide fractional-order approximations for functions with minimal regularity. The resulting schemes allow for the approximation of functions in Cγ[0,T], where 0<γ≤5. A mild invertibility criterion is provided for the implicit schemes. Consistency and stability are proven separately for the whole-number-order approximations and the fractional-order approximations. The rate of convergence in the time variable is shown to be O(τγ), 0<γ≤5 for u∈Cγ[0,T], where τ is the size of the partition of the time mesh. Crucially, the assumption of the integral kernel K being decreasing is not required for the scheme to converge in second-order and below approximations. Optimal convergence results are then proven for both sets of approximations, where fractional-order approximations can obtain up to whole-number rate of convergence in certain scenarios. Finally, numerical examples are provided that illustrate our findings.

About convergence of difference schemes for a third-order pseudo-parabolic equation with nonlocal boundary value condition

About convergence of difference schemes for a third-order pseudo-parabolic equation with nonlocal boundary value condition

Fractional order convergence rate estimate of finite-difference method for the heat equation with concentrated capacity

The convergence of difference scheme for initial-boundary value problem for the heat equation with concentrated capacity and time-dependent coefficient of the space derivatives, is considered. Fractional order convergence rate estimate in a special discrete Sobolev norms, compatible with the smoothness of the coefficient and solution, is proved.

A second order convergent difference scheme for the initial‐boundary value problem of Korteweg–de Vires equation

AbstractIn this article, we present a two‐level implicit difference scheme for Korteweg–de Vires equation with the initial and boundary conditions by the method of order reduction. The truncation error of the difference scheme is analyzed in detail. In the practical computation, the introduced intermediate variable is decoupled in order to reduce the computational cost. It is proved that the difference scheme is solvable by the Browder theorem. The conservation, boundedness, and the unconditional convergence of the numerical solution are also analyzed at length. The convergence order is two both in space and in time in L2‐norm. The numerical solution is proved to be unique under the optimal step ratio condition. Numerical examples demonstrate that the theoretical analysis is correct.

Deciphering disparities in childhood stunting in an underdeveloped state of India: an investigation applying the unconditional quantile regression method

BackgroundUnacceptably high rate of childhood stunting for decades remained a puzzle in the eastern Indian state of Bihar. Despite various programmatic interventions, nearly half of the under-five children (numerically about 10 million) are still stunted in this resource-constrained state.Data and methodsUsing four successive rounds of National Family Health Survey (NFHS) data spread over more than two decades and by employing unconditional quantile regressions and counterfactual decomposition (QR-CD), the present study aims to assess effects of various endowments as well as returns to those endowments in disparities in childhood stunting over the period.ResultsThe results show that although the child’s height-for-age Z-scores (HAZ) disparity largely accounted for differing levels of endowments during the earlier decades, in the later periods, inadequate access to the benefits from various development programmes was also found responsible for HAZ disparities. Moreover, effects of endowments and their returns varied across quantiles. We argue that apart from equalizing endowments, ensuring adequate access to different nutrition-centric programmes is essential to lessen the burden of childhood stunting.ConclusionThe state must focus on intersectoral convergence of different schemes in the form of state nutrition mission, and, strengthen nutrition-centric policy processes and their political underpinnings to harness better dividend.

On the stability and convergence of difference schemes for the generalized fractional diffusion equation with Robin boundary value conditions

On the stability and convergence of difference schemes for the generalized fractional diffusion equation with Robin boundary value conditions

Two-grid algorithm for a system of singularly perturbed reaction-diffusion equations on Shishkin mesh

A system of two coupled singularly perturbed reaction-diffusion equations is considered. A cascadic two-grid algorithm of high accuracy based on a uniform with respect to small parameters convergent difference scheme for such problem has been developed. To increase the accuracy of the difference solution by an order, the Richardson extrapolation method was applied. The results of some numerical experiments are discussed.

M-MATRICES AND CONVERGENCE OF FINITE DIFFERENCE SCHEME FOR PARABOLIC EQUATION WITH INTEGRAL BOUNDARY CONDITION

In the paper, the stability and convergence of difference schemes approximating semilinear parabolic equation with a nonlocal condition are considered. The proof is based on the properties of M-matrices, not requiring the symmetry or diagonal predominance of difference problem. The main presumption is that all the eigenvalues of the corresponding difference problem with nonlocal conditions are positive.

Analysis and application of a convergent difference scheme to nonlinear transport in a Brinkman flow

Purpose The purpose of this paper is to formulate and analyse a convergent numerical scheme and apply it to investigate the coupled problem of fluid flow with heat and mass transfer in a porous channel with variable transport properties. Design/methodology/approach This paper derives the model by assuming a fully developed Brinkman flow with temperature-dependent viscosity and incorporating viscous dissipation, variable transport properties and nonlinear heat and mass sources. For the numerical formulation, the nonlinear sources are treated in semi-implicit manner, whereas the non-constant transport properties are treated by lagging in time leading to decoupled diagonally dominant systems. The consistency, stability and convergence results are derived. The method of manufactured solutions is adopted to numerically verify the theoretical results. The scheme is then applied to investigate the impact of relevant parameters, such as the viscosity parameter, on the flow. Findings Based on the numerical findings, the proposed scheme was found to be unconditionally stable and convergent with first- and second-order accuracy in time and space, respectively. Physical results showed that the flow parameters have influence on the flow fields, particularly, the flow is enhanced by increasing porosity and viscosity parameters and the concentration decreases with increasing diffusivity, whereas both the temperature and Nusselt number decrease with increasing thermal conductivity. Practical implications Numerically, the proposed numerical scheme can be applied without concerns on time steps size restrictions. Non-physical solutions cannot be computed. Physically, the flow can be increased by increasing the viscosity parameters. Pollutants with higher diffusivity will have their concentration decreased faster than those of lower diffusivity. The fluid temperature would decrease faster if its thermal conductivity is higher. Originality/value A fully coupled fluid flow with heat and mass transfer problem having nonlinear properties and nonlinear fractional sources and sink terms, presumably, has not been investigated in a general form as done in this study. The detailed numerical analysis of this particular scheme for the identified general model has also not been considered in the past, to the best of the author’s knowledge.

On Convergence of Difference Schemes for Dirichlet IBVP for Two-Dimensional Quasilinear Parabolic Equations with Mixed Derivatives and Generalized Solutions

Abstract For Dirichlet initial boundary value problem (IBVP) for two-dimensional quasilinear parabolic equations with mixed derivatives monotone linearized difference scheme is constructed. The ellipticity conditions c 1 ⁢ ∑ α = 1 2 ξ α 2 ≤ ∑ α , β = 1 2 k α ⁢ β ⁢ ( u ) ⁢ ξ α ⁢ ξ β ≤ c 2 ⁢ ∑ α = 1 2 ξ α 2 c_{1}\sum_{\alpha=1}^{2}\xi_{\alpha}^{2}\leq\sum_{\alpha,\beta=1}^{2}k_{\alpha% \beta}(u)\xi_{\alpha}\xi_{\beta}\leq c_{2}\sum_{\alpha=1}^{2}\xi_{\alpha}^{2} are assumed to be fulfilled for the sign alternating solution u ⁢ ( 𝐱 , t ) ∈ D ¯ ⁢ ( u ) {u(\mathbf{x},t)\in\bar{D}(u)} only in the domain of exact solution values (unbounded nonlinearity). On the basis of the proved new corollaries of the maximum principle, not only two-sided estimates for the approximate solution y but also its belonging to the domain of exact solution values are established. We assume that the solution is continuous and its first derivatives ∂ ⁡ u ∂ ⁡ x i {\frac{\partial u}{\partial x_{i}}} have discontinuities of the first kind in the neighborhood of the finite number of discontinuity lines. No smoothness of the time derivative is assumed. The convergence of an approximate solution to a generalized solution of a differential problem in the grid norm L 2 {L_{2}} is proved.

Difference schemes for quasi-linear parabolic equations with mixed derivatives

The present paper is devoted to constructing second-order monotone difference schemes for two-dimensional quasi-linear parabolic equation with mixed derivatives. Two-sided estimates of the solution of specific difference schemes for the original problem are obtained, which are fully consistent with similar estimates of the solution of the differential problem, and the a priori estimate in the uniform norm of C is proved. The estimates obtained are used to prove the convergence of difference schemes in the grid norm of L2.

Difference Schemes on Uniform Grids for an Initial-Boundary Value Problem for a Singularly Perturbed Parabolic Convection-Diffusion Equation

Abstract The convergence of difference schemes on uniform grids for an initial-boundary value problem for a singularly perturbed parabolic convection-diffusion equation is studied; the highest x-derivative in the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval ( 0 , 1 ] {(0,1]} . For small ε, the problem involves a boundary layer of width 𝒪 ⁢ ( ε ) {\mathcal{O}(\varepsilon)} , where the solution changes by a finite value, while its derivative grows unboundedly as ε tends to zero. We construct a standard difference scheme on uniform meshes based on the classical monotone grid approximation (upwind approximation of the first-order derivatives). Using a priori estimates, we show that such a scheme converges as { ε ⁢ N } , N 0 → ∞ {\{\varepsilon N\},N_{0}\to\infty} in the maximum norm with first-order accuracy in { ε ⁢ N } {\{\varepsilon N\}} and N 0 {N_{0}} ; as N , N 0 → ∞ {N,N_{0}\to\infty} , the convergence is conditional with respect to N, where N + 1 {N+1} and N 0 + 1 {N_{0}+1} are the numbers of mesh points in x and t, respectively. We develop an improved difference scheme on uniform meshes using the grid approximation of the first x-derivative in the convective term by the central difference operator under the condition h ≤ m ⁢ ε {h\leq m\varepsilon} , which ensures the monotonicity of the scheme; here m is some rather small positive constant. It is proved that this scheme converges in the maximum norm at a rate of 𝒪 ⁢ ( ε - 2 ⁢ N - 2 + N 0 - 1 ) {\mathcal{O}(\varepsilon^{-2}N^{-2}+N^{-1}_{0})} . We compare the convergence rate of the developed scheme with the known Samarskii scheme for a regular problem. It is found that the improved scheme (for ε = 1 {\varepsilon=1} ), as well as the Samarskii scheme, converges in the maximum norm with second-order accuracy in x and first-order accuracy in t.