对流扩散方程的高效稳定差分格式

基于二阶修正Dennis格式 ,提出了采用时间相关法求解定常对流扩散方程的一种具有节省内存空间和提高定常解收敛速度的有理式型优化半隐和松驰半隐紧致格式 .本文建立的差分格式具有运算量小、无网格雷诺数限制的优点 ,是无条件稳定和无条件单调的。通过对非线性Burgers方程进行的数值计算结果表明 ,文中构造的有理式型优化半隐和松驰半隐紧致格式适合于非线性问题计算 ,且保持了无条件稳定和无条件单调的特性 ,尤其能使定常解收敛速度加快 ,精度提高 .     

Flux-splitting schemes for parabolic equations with mixed derivatives

Difference schemes of required quality are often difficult to construct as applied to boundary value problems for parabolic equations with mixed derivatives. Specifically, difficulties arise in the design of monotone difference schemes and unconditionally stable locally one-dimensional splitting schemes. In parabolic problems, certain opportunities are offered by restating the problem in question so that the quantities to be determined are fluxes (directional derivatives). The original problem is then rewritten as a boundary value one for a system of equations in flux variables. Weighted schemes for parabolic equations in flux coordinates are examined. Unconditionally stable locally one-dimensional flux schemes that are first- and second-order accurate in time are constructed for a parabolic equation without mixed derivatives. A feature of systems in flux variables for equations with mixed derivatives is that the terms with time derivatives are coupled with each other.     

Efficient splitting schemes for magneto-hydrodynamic equations

We present in this paper several efficient numerical schemes for the magneto-hydrodynamic (MHD) equations. These semi-discretized (in time) schemes are based on the standard and rotational pressure-correction schemes for the Navier-Stokes equations and do not involve a projection step for the magnetic field. We show that these schemes are unconditionally energy stable, present an effective algorithm for their fully discrete versions and carry out demonstrative numerical experiments.     

构造高分辨率格式的反扩散混合法

In this paper we study a kind of mixed anti-diffusion method for partial differntial equations. Firstly, we use the method to construct some difference schemes for the conservation laws. The schemes are of second order accuracy and are total variation decreasing (TVD). In particular, there are only three knots involved in the schemes. Secondly, we extend the method to construct a few high accuracy difference schemes for elliptic and parabolic equations. Numerical experiments are carried out to illustrate the efficiency of the method.     

Monotonicity criteria for difference schemes designed for hyperbolic equations

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S.K. Godunov’s, A. Harten’s (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils. The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point) stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives. The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic equations.     

A TVD Type Wavelet-Galerkin Method for Hamilton-Jacobi Equations

In this paper,we use Daubechies scaling functions as test functions for the Galerkin method,and discuss Wavelet-Galerkin solutions for the Hamilton-Jacobi equations.It can be proved that the schemesare TVD schemes.Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations.Furthermore,they have high-order accuracy in smooth regions and good resolution of singularities.     

Additive schemes for certain operator-differential equations

Unconditionally stable finite difference schemes for the time approximation of first-order operator-differential systems with self-adjoint operators are constructed. Such systems arise in many applied problems, for example, in connection with nonstationary problems for the system of Stokes (Navier-Stokes) equations. Stability conditions in the corresponding Hilbert spaces for two-level weighted operator-difference schemes are obtained. Additive (splitting) schemes are proposed that involve the solution of simple problems at each time step. The results are used to construct splitting schemes with respect to spatial variables for nonstationary Navier-Stokes equations for incompressible fluid. The capabilities of additive schemes are illustrated using a two-dimensional model problem as an example.     

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.     

On Error Bounds for Approximation Schemes for Non-Convex Degenerate Elliptic Equations

In this paper we provide estimates of the rates of convergence of monotone approximation schemes for non-convex equations in one space-dimension. The equations under consideration are the degenerate elliptic Isaacs equations with x-depending coefficients, and the results applies in particular to certain finite difference methods and control schemes based on the dynamic programming principle. Recently, Krylov, Barles, and Jakobsen obtained similar estimates for convex Hamilton–Jacobi–Bellman equations in arbitrary space-dimensions. Our results are only valid in one space-dimension, but they are the first results of this type for non-convex second-order equations.     

New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations

Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two- and three-dimensions are developed and analyzed. Different from a few sixth-order compact finite difference schemes in the literature, the finite difference and weight coefficients of the new methods have analytic simple expressions. One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points near the boundary and avoids partial derivatives of the source term. Furthermore, the new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6. The coefficient matrices of the new schemes are $M$-matrices for Helmholtz equations with wave number $K≤0,$ which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes. Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.     

Explicit-implicit schemes for convection-diffusion-reaction problems

Boundary value problems for time-dependent convection-diffusion-reaction equations are basic models of problems in continuum mechanics. To study these problems, various numerical methods are used. With a finite difference, finite element, or finite volume approximation in space, we arrive at a Cauchy problem for systems of ordinary differential equations whose operator is asymmetric and indefinite. Explicit-implicit approximations in time are conventionally used to construct splitting schemes in terms of physical processes with separation of convection, diffusion, and reaction processes. In this paper, unconditionally stable schemes for unsteady convection-diffusion-reaction equations are constructed with explicit-implicit approximations used in splitting the operator reaction. The schemes are illustrated by a model 2D problem in a rectangle.     

Monotone compact running schemes for systems of hyperbolic equations

For quasilinear hyperbolic equations, conservative absolutely stable compact schemes are presented that are monotone in a wide range of local Courant numbers. The schemes are fourth-order accurate in space on a compact stencil and first-or third-order accurate in time. They are efficient and are solved by the running calculation method. The convergence rate of the schemes is analyzed in detail in the case of mesh refinement for solutions of various orders of smoothness. The capabilities of the schemes are demonstrated by solving well-known one-dimensional test problems for gas dynamics equations.     

Splitting scheme for poroelasticity and thermoelasticity problems

Boundary value problems in thermoelasticity and poroelasticity (filtration consolidation) are solved numerically. The underlying system of equations consists of the Lamé stationary equations for displacements and nonstationary equations for temperature or pressure in the porous medium. The numerical algorithm is based on a finite-element approximation in space. Standard stability conditions are formulated for two-level schemes with weights. Such schemes are numerically implemented by solving a system of coupled equations for displacements and temperature (pressure). Splitting schemes with respect to physical processes are constructed, in which the transition to a new time level is associated with solving separate elliptic problems for the desired displacements and temperature (pressure). Unconditionally stable additive schemes are constructed by choosing a weight of a three-level scheme.     

Economical discretization schemes for the finite-section method

A problem of economical discretization of equations, arising at the finite-section approximation of Fredholm integral equations of the first kind on the half-line, are investigated. The piecewise-constant discretization schemes providing given accuracy are developed for finite-section equations. The order estimates for the volume of used discrete information are calculated. The comparative analysis of the information expenses for proposed schemes and known earlier is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Family of fifth-order three-level schemes for evolution problems

A multiparameter family of fifth-order three-level schemes in time based on compact approximations is presented for solving evolution problems. The schemes are adapted to hyperbolic and parabolic equations and to stiff systems of ordinary differential equations. In the case of hyperbolic equations, a fifth-order accurate scheme in all variables with compact approximations of spatial derivatives is analyzed. Stability estimates are presented, and the dispersive and dissipative properties are examined.     

Unconditionally stable schemes for convection-diffusion problems

Convection-diffusion problems are basic ones in continuum mechanics. The main features of these problems are connected with the fact that their operators may have an indefinite sign. In this paper we study the stability of difference schemes with weights for convection-diffusion problems where the convective transport operator may have various forms. We construct unconditionally stable schemes for nonstationary convection-diffusion equations based on the use of new variables. Similar schemes are also used for parabolic equations where the operator represents the sum of diffusion operators.     

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.     

Flux-splitting schemes for parabolic problems

Splitting with respect to space variables can be used in solving boundary value problems for second-order parabolic equations. Classical alternating direction methods and locally one-dimensional schemes could be examples of this approach. For problems with rapidly varying coefficients, a convenient tool is the use of fluxes (directional derivatives) as independent variables. The original equation is written as a system in which not only the desired solution but also directional derivatives (fluxes) are unknowns. In this paper, locally one-dimensional additional schemes (splitting schemes) for second-order parabolic equations are examined. By writing the original equation in flux variables, certain two-level locally one-dimensional schemes are derived. The unconditional stability of locally one-dimensional flux schemes of the first and second approximation order with respect to time is proved.     

Fundamental mode exact schemes for unsteady problems

The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for this problem. They are based on a modification of standard discretizations of time derivatives and, in some cases, allow to obtain the exact solution of problems. For multidimensional problems, we can consider the problem of increasing the accuracy only for the most important components of the approximate solution. In the present work, new unconditionally stable schemes for parabolic problems are constructed, which are exact for the fundamental mode. Such two‐level schemes are designed via a modification of standard schemes with weights using Padé approximations. Numerical results obtained for a model problem demonstrate advantages of the proposed fundamental mode exact schemes.     

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.