On some finite difference schemes for solution of hyperbolic heat conduction problems

We consider the accuracy of two finite difference schemes proposed recently in [Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430], and [Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649] to solve an initial-boundary value problem for hyperbolic heat transfer equation. New stability and approximation error estimates are proved and it is noted that some statements given in the above papers should be modified and improved. Finally, two robust finite difference schemes are proposed, that can be used for both, the hyperbolic and parabolic heat transfer equations. Results of numerical experiments are presented.     

Finite difference methods and their physical constraints for the fractional klein‐kramers equation

Incorporating subdiffusive mechanisms into the Klein‐Kramers formalism leads to the fractional Klein‐Kramers equation. Then, the equation can effectively describe subdiffusion in the presence of an external force field in the phase space. This article presents the finite difference methods for numerically solving the fractional Klein‐Kramers equation and does the detailed stability and error analyses. The stability condition, mvβ ≤ 16, shows the ratio between the kinetic energy of the particle and the temperature of the fluid can not be too large, which well agrees with the physical property of the subdiffusive particle, we call it “physical constraint.” The numerical examples are provided to verify the theoretical results on rate of convergence. Moreover, we simulate the fractional Klein‐Kramers dynamics and the simulation results further confirm the effectiveness of our numerical schemes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1561–1583, 2010     

Discontinuous Sturm-Liouville Problems Containing Eigenparameter in the Boundary Conditions

In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. We modify some techniques of [Fulton, C. T., Proc. Roy. Soc. Edin. 77 （A）, 293-308 （1977）], [Walter, J., Math. Z., 133, 301-312 （1973）] and [Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.     

Statistical algorithms for solving the Cauchy problem for second-order parabolic equations: The “dual” scheme

This paper is a continuation of [A. S. Sipin, “Statistical Algorithms for Solving the Cauchy Problem for Second-Order Parabolic Equations,” Vestn. S.-Peterburg. Univ., Mat. Mekh. Astron., No. 3, 65–74 (2011)]. A new algorithm of the Monte Carlo method for solving the Cauchy problem for a second-order parabolic equation with smooth coefficients is considered. Unbiased estimators for functionals of the solutions of this problem are constructed. Unlike in the paper cited above, the “dual” scheme of constructing unbiased estimators for functionals of the solutions of an integral equation equivalent to the Cauchy problem is considered. This simplifies the modeling procedure, because the boundaries of the spectrum for the matrix of the leading coefficients in the equation are not required to be known.     

Uniqueness of selfdual periodic Chern–Simons vortices of topological-type

In analogy with the abelian Maxwell–Higgs model (cf. Jaffe and Taubes in Vortices and monopoles, 1980) we prove that periodic topological-type selfdual vortex-solutions for the Chern–Simons model of Jackiw–Weinberg [Phys Rev Lett 64:2334–2337, 1990] and Hong et al. Phys Rev Lett 64:2230–2233, 1990 are uniquely determined by the location of their vortex points, when the Chern–Simons coupling parameter is sufficiently small. This result follows by a uniqueness and uniform invertibility property established for a related elliptic problem (see Theorem 3.6 and 3.7). Research supported by M.I.U.R. project: Variational Methods and Nonlinear Differential Equations.     

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

The Petrov–Galerkin method for second kind integral equations II: multiwavelet schemes

This paper continues the theme of the recent work [Z. Chen and Y. Xu, The Petrov–Galerkin and iterated Petrov–Galerkin methods for second kind integral equations, SIAM J. Numer. Anal., to appear] and further develops the Petrov–Galerkin method for Fredholm integral equations of the second kind. Specifically, we study wavelet Petrov–Galerkin schemes based on discontinuous orthogonal multiwavelets and prove that the condition number of the coefficient matrix for the linear system obtained from the wavelet Petrov–Galerkin scheme is bounded. In addition, we propose a truncation strategy which forms a basis for fast wavelet algorithms and analyze the order of convergence and computational complexity of these algorithms. This revised version was published online in June 2006 with corrections to the Cover Date.     

Joint estimates of quasi-integrals of three-dimensional skew-symmetric linear differential systems

We prove that the estimates earlier obtained by the authors for the deviations of quasi-integrals of a nonstationary three-dimensional linear differential system with continuously differentiable skew-symmetric coefficient matrix from the corresponding integrals in the stationary case, which are attainable on different nontrivial solutions of the system, are not simultaneously attainable on any single nontrivial solution and at the same time. In the course of proof, we establish new estimates, which are more accurate than those obtained earlier in Mem. Differential Equations Math. Phys., 2007, vol. 41, pp. 157–162. The suggested method for constructing such estimates is also of interest.     

Grid approximations of the pseudospectral type with an improved condition number for second-order differential operators

Schemes that are free of the disadvantages of both the finite-difference and finite-element methods and retain the advantages of the saturation-free grid methods are proposed and investigated. The asymptotic behavior of their maximal N th eigenvalue is the same as the behavior of the N the eigenvalue of a differential operator, and it is not difficult to apply these discretizations to nonstationary problems. In contrast to polynomial pseudospectral approximations, the schemes of this paper, as well as of E. B. Karpilovskaya, “Convergence of the collocation method,” Sov. Math. Dokl.,4, No. 2, 1070–1073 (1963) and I. P. Gavrilyuk and L. D. Grekov, On Algorithms for the Realization of Grid Schemes without the Accuracy Staturation for Second-Order Ordinary Differential Equations [in Russian], Deposited at UkrNIINTI 16.08.1991, utilize uniform grids. Bibliography: 27 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 1–27.     

On the convergence of difference schemes for the diffusion equation of fractional order

For the diffusion equation of fractional order, we construct an approximation difference scheme of order 0(h ² + τ). We present an algorithm for the solution of boundary-value problems for a generalized transfer equation of fractional order. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 994–996, July, 1998.     

On the convergence of Newton-type methods under mild differentiability conditions

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.     

A residual based error estimator for the Mimetic Finite Difference method

We present a local error indicator for the Mimetic Finite Difference method for diffusion-type problems on polyhedral meshes. Under essentially the same general hypotheses used in (SIAM J. Numer. Anal. 43:1872–1896, 2005) to show the convergence of the method, we prove the global reliability and local efficiency of the proposed estimator.     

Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).     

A high‐order compact scheme for the nonlinear fractional Klein–Gordon equation

In this article, a high‐order finite difference scheme for a kind of nonlinear fractional Klein–Gordon equation is derived. The time fractional derivative is described in the Caputo sense. The solvability of the difference system is discussed by the Leray–Schauder fixed point theorem, while the stability and L_∞ convergence of the finite difference scheme are proved by the energy method. Numerical examples are provided to demonstrate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 706–722, 2015     

Error estimates for a stabilized finite element method for the Oldroyd B model

In this paper, we study a new approximation scheme of transient viscoelastic fluid flow obeying an Oldroyd-B-type constitutive equation. The new stabilized formulation bases on the choice of a modified Euler method connected to the streamline upwinding Petrov-Galerkin (SUPG) method [M. Bensaada, D. Esselaoui, D. Sandri, Stabilization method for continuous approximation of transient convection problem, Numer. Methods Partial Differential Equations 21 (2004) 170-189], in order to stabilize the tensorial transport term of the Oldroyd derivative. Suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution. A priori error estimates for the approximation in terms of the mesh parameter h and the time discretization parameter Δt are derived.     

Mixed problem for a nonlinear ultraparabolic equation that generalizes the diffusion equation with inertia

We consider a mixed problem for a nonlinear ultraparabolic equation that is a nonlinear generalization of the diffusion equation with inertia and the special cases of which are the Fokker-Planck equation and the Kolmogorov equation. Conditions for the existence and uniqueness of a solution of this problem are established. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1192–1210, September, 2006.     

Superconvergence in the generalized finite element method

In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945–981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727–758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289–314, 1996). In particular, we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct. I. Babuška’s research was partially supported by NSF Grant # DMS-0341982 and ONR Grant # N00014-99-1-0724. U. Banerjee’s research was partially supported by NSF Grant # DMS-0341899. J. E. Osborn’s research was supported by NSF Grant # DMS-0341982.     

A linearized high‐order difference scheme for the fractional Ginzburg–Landau equation

The numerical solution for the one‐dimensional complex fractional Ginzburg–Landau equation is considered and a linearized high‐order accurate difference scheme is derived. The fractional centered difference formula, combining the compact technique, is applied to discretize fractional Laplacian, while Crank–Nicolson/leap‐frog scheme is used to deal with the temporal discretization. A rigorous analysis of the difference scheme is carried out by the discrete energy method. It is proved that the difference scheme is uniquely solvable and unconditionally convergent, in discrete maximum norm, with the convergence order of two in time and four in space, respectively. Numerical simulations are given to show the efficiency and accuracy of the scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 105–124, 2017     

On Constrained Nonlinear Hermite Subdivision

We determine shape-preserving regions and we describe a general setting to generate shape-preserving families for the 2-points Hermite subdivision scheme introduced by Merrien (Numer. Algorithms 2:187–200, [1992]). This general construction includes the shape-preserving families presented in Merrien and Sablonníere (Constr. Approx. 19:279–298, [2003]) and Pelosi and Sablonníere (C ¹ GP Hermite Interpolants Generated by a Subdivision Scheme, Prépublication IRMAR 06–23, Rennes, [2006]). New special families are presented as particular examples. Nonstationary and nonuniform versions of such schemes, which produce smoother limits, are discussed.      

Analysis of the fractional Kawahara equation using an implicit fully discrete local discontinuous Galerkin method

In this article, an implicit fully discrete local discontinuous Galerkin (LDG) finite element method, on the basis of finite difference method in time and LDG method in space, is applied to solve the time‐fractional Kawahara equation, which is introduced by replacing the integer‐order time derivatives with fractional derivatives. We prove that our scheme is unconditional stable and convergent through analysis. Extensive numerical results are provided to demonstrate the performance of the present method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013