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.     

波动方程两种哈密顿型蛙跳格式

1.构造格式 考虑如下波动方程 u_(tt)=u_(xx) (1.1)的初边值问题,设其边界条件为周期的,即在此条件下,解具有周期性.(1.1)有二种namilton形式.一种是经典形式:     

Construction of splitting schemes based on transition operator approximation

The stability analysis of approximate solutions to unsteady problems for partial differential equations is usually based on the use of the canonical form of operator-difference schemes. Another possibility widely used in the analysis of methods for solving Cauchy problems for systems of ordinary differential equations is associated with the estimation of the norm of the transition operator from the current time level to a new one. The stability of operator-difference schemes for a first-order model operator-differential equation is discussed. Primary attention is given to the construction of additive schemes (splitting schemes) based on approximations of the transition operator. Specifically, classical factorized schemes, componentwise splitting schemes, and regularized operator-difference schemes are related to the use of a certain multiplicative transition operator. Additive averaged operator-difference schemes are based on an additive representation of the transition operator. The construction of second-order splitting schemes in time is discussed. Inhomogeneous additive operator-difference schemes are constructed in which various types of transition operators are used for individual splitting operators.     

Nonstandard finite difference schemes for a class of generalized convection–diffusion–reaction equations

In this work, a class of nonstandard finite difference (NSFD) schemes are proposed to approximate the solutions of a class of generalized convection–diffusion–reaction equations. First, in the case of no diffusion, two exact finite difference schemes are presented using the method of characteristics. Based on these two exact schemes, a class of exact schemes are presented by introducing a parameter α. Second, since the forms of these exact schemes are so complicated that they are not convenient to use, a class of NSFD schemes are derived from the exact schemes using numerical approximations. It follows that, under certain conditions about denominator function of time‐step sizes, these NSFD schemes are elementary stable and the solutions are positive and bounded. Third, by means of the Mickens' technique of subequations, a new class of implicit NSFD schemes are constructed for the full convection–diffusion–reaction equations. It is shown that, under certain parameters set, these NSFD schemes are capable of preserving the non‐negativity and boundedness of the analytical solutions. Finally, some numerical simulations are provided to verify the validity of our analytical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1288–1309, 2015     

Classification of association schemes with 18 and 19 vertices

The isomorphism classes of association schemes with 18 and 19 vertices are classified. We obtain 95 isomorphism classes of association schemes with 18 vertices and denote the representatives of the isomorphism classes as subschemes of 7 association schemes. We obtain 7 isomorphism classes of association schemes with 19 vertices and six of them are cyclotomic schemes.     

Commutative Association Schemes Whose Symmetrizations Have Two Classes

If a symmetric association scheme of class two is realized as the symmetrization of a commutative association scheme, then it either admits a unique symmetrizable fission scheme of class three or four, or admits three fission schemes, two of which are class three and one is of class four. We investigate the classification problem for symmetrizable (commutative) association schemes of two-class symmetric association schemes. In particular, we give a classification of association schemes whose symmetrizations are obtained from completely multipartite strongly regular graphs in the notion of wreath product of two schemes. Also the cyclotomic schemes associated to Paley graphs and their symmetrizable fission schemes are discussed in terms of their character tables.     

一类交错网格的Gauss型格式

本文在交错网格的情况下 ,利用 Gauss型求积公式构造了一类不需解 Riemann问题的求解一维单个双曲守恒律的二阶显式 Gauss型差分格式 ,证明了该格式在CFL条件限制下为 TVD格式 ,并证明了这类格式的收敛性 ,然后将格式推广到方程组的情形 .由于在交错网格的情况下构造的这类差分格式 ,不需要求解 Riemann问题 ,因此这类格式与诸如 Harten等的 TVD格式相比具有如下优点 :由于不需要完整的特征向量系 ,因此可用于求解弱双曲方程组 ,计算更快、编程更加简便等 .     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

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.     

Discontinuous mixed penalty-free Galerkin method for second-order quasilinear elliptic equations

Discrete schemes for finding an approximate solution of the Dirichlet problem for a second-order quasilinear elliptic equation in conservative form are investigated. The schemes are based on the discontinuous Galerkin method (DG schemes) in a mixed formulation and do not involve internal penalty parameters. Error estimates typical of DG schemes with internal penalty are obtained. A new result in the analysis of the schemes is that they are proved to satisfy the Ladyzhenskaya-Babuska-Brezzi condition (inf-sup) condition.     

Minimal dissipation hybrid bicompact schemes for hyperbolic equations

New monotonicity-preserving hybrid schemes are proposed for multidimensional hyperbolic equations. They are convex combinations of high-order accurate central bicompact schemes and upwind schemes of first-order accuracy in time and space. The weighting coefficients in these combinations depend on the local difference between the solutions produced by the high- and low-order accurate schemes at the current space-time point. The bicompact schemes are third-order accurate in time, while having the fourth order of accuracy and the first difference order in space. At every time level, they can be solved by marching in each spatial variable without using spatial splitting. The upwind schemes have minimal dissipation among all monotone schemes constructed on a minimum space-time stencil. The hybrid schemes constructed has been successfully tested as applied to a number of two-dimensional gas dynamics benchmark problems.     

On compact approximations of divergence differential equations

A method for the construction of compact difference schemes approximating divergence differential equations is proposed. The schemes have an arbitrarily prescribed order of approximation on general stencils. It is shown that the construction of such schemes for partial differential equations is based on special compact schemes approximating ordinary differential equations in several independent functions. Necessary and sufficient conditions on the coefficients of these schemes with high order of approximation are obtained. Examples of reconstruction of compact difference schemes for partial differential equations with these schemes are given. It is shown that such compact difference schemes have the same order of accuracy both for classical approximations on smooth solutions and weak approximations on discontinuous solutions.     

Association schemes related to Delsarte–Goethals codes

In this paper, we construct an infinite series of 9-class association schemes from a refinement of the partition of Delsarte–Goethals codes by their Lee weights. The explicit expressions of the dual schemes are determined through direct manipulations of complicated exponential sums. As a byproduct, another three infinite families of association schemes are also obtained as fusion schemes and quotient schemes.     

Optimal discrete and continuous mono‐implicit Runge–Kutta schemes for BVODEs

Recent investigations of discretization schemes for the efficient numerical solution of boundary value ordinary differential equations (BVODEs) have focused on a subclass of the well‐known implicit Runge–Kutta (RK) schemes, called mono‐implicit RK (MIRK) schemes, which have been employed in two software packages for the numerical solution of BVODEs, called TWPBVP and MIRKDC. The latter package also employs continuous MIRK (CMIRK) schemes to provide C ¹ continuous approximate solutions. The particular schemes implemented in these codes come, in general, from multi‐parameter families and, in some cases, do not represent optimal choices from these families. In this paper, several optimization criteria are identified and applied in the derivation of optimal MIRK and CMIRK schemes for orders 1–6. In some cases the schemes obtained result from the analysis of existent multi‐parameter families; in other cases new families are derived from which specific optimal schemes are then obtained. New MIRK and CMIRK schemes are presented which are superior to those currently available. Numerical examples are provided to demonstrate the practical improvements that can be obtained by employing the optimal schemes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Fitting fat curves

Four schemes for fitting fat curves are considered. Two of the schemes fit fat curves, assuming that the curvature of the axis curve is specified and two of the schemes fit fat curves to a set of sample points. In each of the two cases noninterval and interval schemes are discussed.     

Arbitrary-order difference schemes for solving linear advection equations with constant coefficients by the Godunov method with antidiffusion

An approach to the construction of second-and higher order accurate difference schemes in time and space is described for solving the linear one-and multidimensional advection equations with constant coefficients by the Godunov method with antidiffusion. The differential approximations for schemes of up to the fifth order are constructed and written. For multidimensional advection equations with constant coefficients, it is shown that Godunov schemes with splitting over spatial variables are preferable, since they have a smaller truncation error than schemes without splitting. The high resolution and efficiency of the difference schemes are demonstrated using test computations.     

Difference methods for computing the Ginzburg‐Landau equation in two dimensions

In this article, three difference schemes of the Ginzburg‐Landau Equation in two dimensions are presented. In the three schemes, the nonlinear term is discretized such that nonlinear iteration is not needed in computation. The plane wave solution of the equation is studied and the truncation errors of the three schemes are obtained. The three schemes are unconditionally stable. The stability of the two difference schemes is proved by induction method and the time‐splitting method is analysized by linearized analysis. The algebraic multigrid method is used to solve the three large linear systems of the schemes. At last, we compute the plane wave solution and some dynamics of the equation. The numerical results demonstrate that our schemes are reliable and efficient. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011py; 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 507–528, 2011     

Graded group schemes and graded group varieties

We define graded group schemes and graded group varieties and develop their theory. Graded group schemes are the graded analogue of a?ne group schemes and are in correspondence with graded Hopf algebras. Graded group varieties take the place of infinitesimal group schemes. We generalize the result that connected graded bialgebras are graded Hopf algebra to our setting and we describe the algebra structure of graded group varieties. We relate these new objects to the classical ones providing a new and broader framework for the study of graded Hopf algebras and a?ne group schemes.     

Families of univariate and bivariate subdivision schemes originated from quartic B-spline

Families of parameter dependent univariate and bivariate subdivision schemes are presented in this paper. These families are new variants of the Lane-Riesenfeld algorithm. So the subdivision algorithms consist of both refining and smoothing steps. In refining step, we use the quartic B-spline based subdivision schemes. In smoothing step, we average the adjacent points. The bivariate schemes are the non-tensor product version of our univariate schemes. Moreover, for odd and even number of smoothing steps, we get the primal and dual schemes respectively. Higher regularity of the schemes can be achieved by increasing the number of smoothing steps. These schemes can be nicely generalized to contain local shape parameters that allow the user to adjust locally the shape of the limit curve/surface.     

Classification of explicit three-stage symplectic difference schemes for the numerical solution of natural Hamiltonian systems: A comparative study of the accuracy of high-order schemes on molecular dynamics problems

The natural Hamiltonian systems (systems with separable Hamiltonians) are considered. The variety of explicit three-stage symplectic schemes is described. A classification of the third-order accurate schemes is given. All fourth-order schemes are found (there are seven of them). It is proved that there are no fifth-order schemes. The schemes with improved properties, such as invertibility and optimality with respect to the phase error, are listed. Numerical results that demonstrate the properties of these schemes are presented, and their comparative analysis with respect to the accuracy–efficiency criterion is given. The disbalance of total energy is used as the accuracy criterion.