Regularized additive operator-difference schemes

The construction of additive operator-difference (splitting) schemes for the approximate solution Cauchy problem for the first-order evolutionary equation is considered. Unconditionally stable additive schemes are constructed on the basis of the Samarskii regularization principle for operator-difference schemes. In the case of arbitrary multicomponent splitting, these schemes belong to the class of additive full approximation schemes. Regularized additive operator-difference schemes for evolutionary problems are constructed without the assumption that the regularizing operator and the operator of the problem are commutable. Regularized additive schemes with double multiplicative perturbation of the additive terms of the problem’s operator are proposed. The possibility of using factorized multicomponent splitting schemes, which can be used for the approximate solution of steadystate problems (finite difference relaxation schemes) are discussed. Some possibilities of extending the proposed regularized additive schemes to other problems are considered.     

Two-grid discretization schemes for nonlinear Schrödinger equations

We study efficient two-grid discretization schemes with two-loop continuation algorithms for computing wave functions of two-coupled nonlinear Schrödinger equations defined on the unit square and the unit disk. Both linear and quadratic approximations of the operator equations are exploited to derive the schemes. The centered difference approximations, the six-node triangular elements and the Adini elements are used to discretize the PDEs defined on the unit square. The proposed schemes also can compute stationary solutions of parameter-dependent reaction–diffusion systems. Our numerical results show that it is unnecessary to perform quadratic approximations.     

Operator-difference schemes for a class of systems of evolution equations

For a special system of evolution equations of first order, discrete time approximations for the approximate solution of the Cauchy problem are considered. Such problems arise after the spatial approximation in the Schrödinger equation and the subsequent separation of the imaginary and real parts and in nonstationary problems of acoustics and electrodynamics. Unconditionally stable two time level operator-difference weighted schemes are constructed. The second class of difference schemes is based on the formal passage to explicit operator-difference schemes for evolution equations of second order when explicit-implicit approximation is used for isolated equations of the system. The regularization of such schemes in order to obtain unconditionally stable operator difference schemes is discussed. Splitting schemes involving the solution of simplest problems at each time step are constructed.     

Difference schemes for solving the Cauchy problem for a second-order operator differential equation

A class of finite-difference schemes for solving an ill-posed Cauchy problem for a second-order linear differential equation with a sectorial operator in a Banach space is studied. Time-uniform estimates of the convergence rate and the error of such schemes are obtained. Previously known estimates are improved due to an optimal choice of initial data for a difference scheme.     

Bregman operator splitting with variable stepsize for total variation image reconstruction

This paper develops a Bregman operator splitting algorithm with variable stepsize (BOSVS) for solving problems of the form $\min\{\phi(Bu) +1/2\|Au-f\|_{2}^{2}\}$ , where ? may be nonsmooth. The original Bregman Operator Splitting (BOS) algorithm employed a fixed stepsize, while BOSVS uses a line search to achieve better efficiency. These schemes are applicable to total variation (TV)-based image reconstruction. The stepsize rule starts with a Barzilai-Borwein (BB) step, and increases the nominal step until a termination condition is satisfied. The stepsize rule is related to the scheme used in SpaRSA (Sparse Reconstruction by Separable Approximation). Global convergence of the proposed BOSVS algorithm to a solution of the optimization problem is established. BOSVS is compared with other operator splitting schemes using partially parallel magnetic resonance image reconstruction problems. The experimental results indicate that the proposed BOSVS algorithm is more efficient than the BOS algorithm and another split Bregman Barzilai-Borwein algorithm known as SBB.     

Additive Difference Schemes and Iteration Methods for Problems of Mathematical Physics

We construct additive difference schemes for first-order differential–operator equations. The exposition is based on the general theory of stability for operator–difference schemes in lattice Hilbert spaces. The main focus is on the case of additive decomposition with an arbitrary number of mutually noncommuting operator terms. Additive schemes for second-order evolution equations are considered in the same way.     

Discrete models in the dynamic problem of linear elasticity and conservation laws

The dynamic problem of linear elasticity (a continuous model) is based on the momentum and moment of momentum conservation laws and hence has an additional conservation law for the total energy (the kinetic plus potential energy). Continuous models with this property are referred to as entropy models (S.K. Godunov). For difference schemes, this property is described with the use of the notion of complete conservativeness) (A.A. Samarskii). For the considered problem, we make a complete analysis of a two-parameter family of “conjugatecoordinated” two-layer difference schemes including a conservative scheme. For the latter, we construct efficient triangular-factorized implementations with the same parallelism degree as in ordinary explicit schemes. For difference schemes without the property of complete conservativeness, we discuss the role of the law of passage to the limit for the minimization of imbalance in the total energy.     

Monotone schemes for a class of nonlinear elliptic and parabolic problems

We construct monotone numerical schemes for a class of nonlinear PDE for elliptic and initial value problems for parabolic problems. The elliptic part is closely connected to a linear elliptic operator, which we discretize by monotone schemes, and solve the nonlinear problem by iteration. We assume that the elliptic differential operator is in the divergence form, with measurable coefficients satisfying the strict ellipticity condition, and that the right-hand side is a positive Radon measure. The numerical schemes are not derived from finite difference operators approximating differential operators, but rather from a general principle which ensures the convergence of approximate solutions. The main feature of these schemes is that they possess stencils stretching far from basic grid-rectangles, thus leading to system matrices which are related to M-matrices.     

Low dissipative entropy stable schemes using third order WENO and TVD reconstructions

A low dissipative framework is given to construct high order entropy stable flux by addition of suitable numerical diffusion operator into entropy conservative flux. The framework is robust in the sense that it allows the use of high order reconstructions which satisfy the sign property only across the discontinuities. The third order weighted essentially non-oscillatory (WENO) interpolations and high order total variation diminishing (TVD) reconstructions are shown to satisfy the sign property across discontinuities. Third order accurate entropy stable schemes are constructed by using third order WENO and high order TVD reconstructions procedures in the diffusion operator. These schemes are efficient and less diffusive since the diffusion is actuated only in the sign stability region of the used reconstruction which includes discontinuities. Numerical results with constructed schemes for various test problems are given which show the third order accuracy and less dissipative nature of the schemes.     

High-order compact solvers for the three-dimensional Poisson equation

New compact approximation schemes for the Laplace operator of fourth- and sixth-order are proposed. The schemes are based on a Padé approximation of the Taylor expansion for the discretized Laplace operator. The new schemes are compared with other finite difference approximations in several benchmark problems. It is found that the new schemes exhibit a very good performance and are highly accurate. Especially on large grids they outperform noncompact schemes.     

Unconditional and optimal <Emphasis Type="Bold">H</Emphasis><Superscript>2</Superscript>-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions

The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear Klein-Gordon-Schrödinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H ²-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h ² + τ ²) with mesh-size h and time step τ in the discrete H ²-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.     

A combination difference scheme for the eigenvalue problem of the Laplace operator

The paper constructs and analyzes a combination difference scheme for numerical determination of the eigenvalues of the Laplace operator. The proposed scheme uses the two-sided (from above and from below) properties of variational-difference and ordinary difference schemes for the eigenvalue problem of the Laplace operator in convex domains. The half-sum of the two schemes in convex domains gives an O(h⁴) approximation to the exact eigenvalue. A summation representation formula is constructed as an implementation of the ten-point difference scheme.Kiev University. Nukus University. Turkmen Teachers College. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 55–60, 1991.     

Three-level schemes of the alternating triangular method

In this paper, the schemes of the alternating triangular method are set out in the class of splitting methods used for the approximate solution of Cauchy problems for evolutionary problems. These schemes are based on splitting the problem operator into two operators that are conjugate transposes of each other. Economical schemes for the numerical solution of boundary value problems for parabolic equations are designed on the basis of an explicit-implicit splitting of the problem operator. The alternating triangular method is also of interest for the construction of numerical algorithms that solve boundary value problems for systems of partial differential equations and vector systems. The conventional schemes of the alternating triangular method used for first-order evolutionary equations are two-level ones. The approximation properties of such splitting methods can be improved by transiting to three-level schemes. Their construction is based on a general principle for improving the properties of difference schemes, namely, on the regularization principle of A.A. Samarskii. The analysis conducted in this paper is based on the general stability (or correctness) theory of operator-difference schemes.     

Stability of an operator-difference scheme for thermoelasticity problems

We study a linear three-layer operator-difference scheme with weights which generalizes a class of difference and projection-difference schemes for coupled thermoelasticity problems. Using the method of energy inequalities, we obtain stability estimates in grid energy norms under certain conditions on operator coefficients and parameters of the scheme.     

Perturbation theory of difference schemes

Summary This paper is to show, if the abstract Cauchy problem has a stable difference scheme, then the Cauchy problem of a perturbed equation has also a stable difference scheme when a perturbing operator and its difference approximation have some suitable properties. And it will be noted this result is applicable to parabolic differential equations and their lower order terms, when parabolic difference schemes are used as original difference schemes.     

Computing eigenelements of Sturm-Liouville problems by using Daubechies wavelets

This work is our first step to get multiresolution approximation of eigenelements of Sturm-Liouville problems within bounded domain of varied nature. The formula for obtaining elements of representation of Sturm-Liouville operator involving polynomial coefficients in wavelet basis of Daubechies family have been derived in a form which can be readily used for their computations by a simple computer program. Estimates of errors for both the eigenvalues and eigenfunctions are also presented here. The proposed wavelet-Galerkin scheme based on scale functions and wavelets of Daubechies family having three or four vanishing moments of their wavelets has been applied to get approximate eigenelements of regular and singular Sturm-Liouville problems within bounded domain and compared with the exact or approximate results whenever available. From our study it appears that the proposed method is efficient and rapidly convergent in comparison to other approximation schemes based on variational method in Haar basis or finite difference methods studied by Bujurke et al. [39].     

Conservative compact difference schemes for the coupled nonlinear schrödinger system

In this article, some conservative compact difference schemes are explored for the strongly coupled nonlinear schrödinger system. After transforming the scheme into matrix form, we prove the existence and uniqueness, convergence and stability of the difference solutions for one nonlinear scheme in the norm by using some techniques of matrix theory. Numerical results show that one nonlinear scheme is the most efficient of all the compact schemes constructed here. It allows much larger time steps than the others. The second most efficient compact scheme is a linear one. We then give numerical simulations to two soliton interactions for the two most efficient compact schemes. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 749–772, 2014     

Efficient revocable identity-based encryption via subset difference methods

Providing an efficient revocation mechanism for identity-based encryption (IBE) is very important since a user’s credential (or private key) can be expired or revealed. revocable IBE (RIBE) is an extension of IBE that provides an efficient revocation mechanism. Previous RIBE schemes essentially use the complete subtree (CS) scheme of Naor, Naor and Lotspiech (CRYPTO 2001) for key revocation. In this paper, we present a new technique for RIBE that uses the efficient subset difference (SD) scheme of Naor et al. instead of using the CS scheme to improve the size of update keys. Following our new technique, we first propose an efficient RIBE scheme in prime-order bilinear groups by combining the IBE scheme of Boneh and Boyen and the SD scheme and prove its selective security under the standard assumption. Our RIBE scheme is the first RIBE scheme in bilinear groups that has O(r) number of group elements in an update key where r is the number of revoked users. Next, we also propose another RIBE scheme in composite-order bilinear groups and prove its full security under static assumptions. Our RIBE schemes also can be integrated with the layered subset difference scheme of Halevy and Shamir (CRYPTO 2002) to reduce the size of a private key.     

High‐order compact difference schemes for the modified anomalous subdiffusion equation

In this article, two kinds of high‐order compact finite difference schemes for second‐order derivative are developed. Then a second‐order numerical scheme for a Riemann–Liouvile derivative is established based on a fractional centered difference operator. We apply these methods to a fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability, and convergence analysis of these difference schemes are studied by using Fourier method. The convergence orders of these numerical schemes are and , respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 213–242, 2016