On the multioperator method for constructing approximations and finite difference schemes of an arbitrarily high order

Main results of the development of the multioperator method for constructing approximations of prescribed order are presented. Multioperators for various approximation problems are considered. The focus is on the multioperators for convective terms in fluid dynamics equations. Types of multioperator schemes are described and possibilities for their optimization are discussed. Results of solving benchmark problems in the case of tenth- and 18th-order schemes are presented.     

A parallel computational scheme with ninth-order multioperator approximations and its application to direct numerical simulation

The properties of ninth-order multioperator compact schemes based on known third-and fifth-order compact approximations are examined. The domains where the multioperators have fixed signs are determined numerically. The numerical results are compared with the exact solution to the Burgers equation. The multioperator schemes are applied to the problem of vortex sheet roll-up.     

Multioperator representation of composite compact schemes

The multioperator approach is used to obtain high-order accurate compact differences. These differences are developed to describe convective terms of differential equations, as well as mixed derivatives, source terms, and the coefficients of metric derivatives of coordinate transformations. The same principles are used to obtain high-order compact differences for representing diffusion terms. These differences underlie multioperator composite compact schemes, which are used to compute the flow past an airfoil by integrating the nonstationary Navier-Stokes equations supplemented with the equations of a turbulent viscosity model.     

On the use of multioperators in the construction of high-order grid approximations

Examples of using the multioperator technique in order to increase the order of accuracy of some linear operators are given. Formulas for numerical differentiation, approximation of diffusion terms, recalculation, filtration, and extrapolation of grid functions are considered. A new family of multioperator approximations for convective terms of equations is presented. Multioperators of 16th and 32nd orders are analyzed as an example.     

On families high-order accurate multioperator approximations of derivatives using two-point operators

A new family of multioperator approximations to derivatives of even and odd orders with inversion of two-point operators is considered. Existence and uniqueness theorems are stated for multioperators of formally arbitrary orders, and their spectral properties are examined. A scheme for a test hyperbolic equation with a multioperator approximation of 36th order is analyzed as an example. The accuracy and convergence of numerical solutions to the test problem are estimated.     

Tenth-order accurate multioperator scheme and its application in direct numerical simulation

A tenth-order accurate multioperator difference scheme based on two-point compact noncentered operators is described. Optimal sets of parameters ensuring the smallness of the phase and amplitude errors are found. Results obtained by the numerical simulation of the instability of a hot subsonic jet are discussed. Characteristics of excited acoustic fields are presented.     

Operads and varieties of algebras defined by polylinear identities

We show that varieties of algebras over abstract clones and over the corresponding operads are rationally equivalent. We introduce the class of operads (which we call commutative for definiteness) such that the varieties of algebras over these operads resemble in a sense categories of modules over commutative rings. In particular, the notions of a polylinear mapping and the tensor product of algebras. The categories of modules over commutative rings and the category of convexors are examples of varieties over commutative operads. By analogy with the theory of linear multioperator algebras, we develop a theory of C-linear multioperator algebras; in particular, of algebras, defined by C-polylinear identities (here C is a commutative operad). We introduce and study symmetric C-linear operads. The main result of this article is as follows: A variety of C-linear multioperator algebras is defined by C-polylinear identities if and only if it is rationally equivalent to a variety of algebras over a symmetric C-linear operad.     

Guiding functions and periodic solutions for inclusions with causal multioperators

In the present paper, the method of guiding functions is applied to study the periodic problem for a differential inclusion with a causal multioperator. At first we consider the case when the multioperator is closed and convex-valued. Then the case of a non-convex-valued and lower semicontinuous right-hand part is considered. Thereafter, the theory is extended to the case of non-smooth guiding functions.     

Superalgebras and operads. I

We construct a theory of multioperator superalgebras and superalgebras over operad.     

Ω-Foliated Formations of Multioperator <Emphasis Type="Italic">T</Emphasis>-Groups

We define and construct several types of Ω-foliated formations of multioperator T-groups with composition series. We describe the structure of minimal and complete satellites and apply them to study the properties of lattices and products of these formations.     

Numerical simulation of shear layer instability using a scheme with ninth-order multioperator approximations

For equations with convective terms, a difference scheme is described based on ninth-order multioperator approximations. Its optimization aimed at achieving a high resolution of small scales of solutions is discussed. The scheme is applied to test problems, and shear layer instability is numerically simulated with a detailed analysis of developing vortex structures and their characteristics.     

Application of compact and multioperator schemes to the numerical simulation of acoustic fields generated by instability waves in supersonic jets

Acoustic fields generated by instability waves in supersonic jets were numerically simulated. A seventh-order multioperator scheme was used to solve the Euler equations linearized about the mean flow field in an axisymmetric turbulent jet. The mean field was computed using fifth-order compact approximations of the convective terms under conditions similar to experimental data. The numerical results were found to agree well with the experiment.     

On superalgebras over operads

We show that the class of varieties of multioperator superalgebras, defined by multilinear identities and considered up to rational equivalence, coincides with the class of varieties of superalgebras over an operad. We define the concepts of Grassmann and Clifford envelopes in the most general case and study their properties. We define the concept of module over a superalgebra over an operad and the concept of universal enveloping superalgebra for an algebra over an operad and study their properties.     

Conservative semi‐Lagrangian finite volume schemes

Semi‐Lagrangian finite volume schemes for the numerical approximation of linear advection equations are presented. These schemes are constructed so that the conservation properties are preserved by the numerical approximation. This is achieved using an interpolation procedure based on area‐weighting. Numerical results are presented illustrating some of the features of these schemes. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:403–425, 2001     

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     

Centred TVD schemes for hyperbolic conservation laws

New first- and high-order centred methods for conservation lawsare presented. Convenient TVD conditions for constructing centredTVD schemes are then formulated and some useful results areproved. Two families of centred TVD schemes are constructedand extended to nonlinear systems. Some numerical results arealso presented.     

A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise

The paper deals with the numerical treatment of stochastic differential-algebraic equations of index one with a scalar driving Wiener process. Therefore, a particularly customized stochastic Runge-Kutta method is introduced. Order conditions for convergence with order 1.0 in the mean-square sense are calculated and coefficients for some schemes are presented. The proposed schemes are stiffly accurate and applicable to nonlinear stochastic differential-algebraic equations. As an advantage they do not require the calculation of any pseudo-inverses or projectors. Further, the mean-square stability of the proposed schemes is analyzed and simulation results are presented bringing out their good performance.     

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 .     

The third-order relaxation schemes for hyperbolic conservation laws

A class of semi-discrete third-order relaxation schemes are presented for relaxation systems which approximate systems of hyperbolic conservation laws. These schemes for the scalar conservation law are shown to satisfy the property of total variation diminishing (TVD) in the zero relaxation limit. A third-order TVD Runge–Kutta splitting method is developed for the temporal discretization of the semi-discrete schemes. Numerical results are given illustrating these schemes on one-dimensional nonlinear problems.     

Nonlinear Implicit One-Step Schemes for Solving Initial Value Problems for Ordinary Differential Equation with Steep Gradients

A general theory for nonlinear implicit one-step schemes for solving initial value problems for ordinary differential equations is presented in this paper. The general expansion of "symmetric" implicit one-step schemes having second-order is derived and stability and convergence are studied. As examples, some geometric schemes are given. Based on previous work of the first author on a generalization of means, a fourth-order nonlinear implicit one-step scheme is presented for solving equations with steep gradients. Also, a hybrid method based on the GMS and a fourth-order linear scheme is discussed. Some numerical results are given.