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1.
P. N. Vabishchevich 《Computational Mathematics and Mathematical Physics》2010,50(3):428-436
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. 相似文献
2.
波动方程两种哈密顿型蛙跳格式 总被引:15,自引:3,他引:15
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: 相似文献
3.
P. N. Vabishchevich 《Computational Mathematics and Mathematical Physics》2012,52(2):235-244
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. 相似文献
4.
V. V. Ostapenko 《Numerical Analysis and Applications》2012,5(3):242-253
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. 相似文献
5.
P. N. Vabishchevich 《Computational Mathematics and Mathematical Physics》2010,50(12):2033-2043
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. 相似文献
6.
N. Ya. Moiseev I. Yu. Silant’eva 《Computational Mathematics and Mathematical Physics》2008,48(7):1210-1220
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. 相似文献
7.
P. N. Vabishchevich 《Computational Mathematics and Mathematical Physics》2012,52(6):887-894
The spectral mimetic (SM) properties of operator-difference schemes for solving the Cauchy problem for first-order evolutionary equations concern the time evolution of individual harmonics of the solution. Keeping track of the spectral characteristics makes it possible to select more appropriate approximations with respect to time. Among two-level implicit schemes of improved accuracy based on Padé approximations, SM-stability holds for schemes based on polynomial approximations if the operator in an evolutionary equation is self-adjoint and for symmetric schemes if the operator is skew-symmetric. In this paper, additive schemes (also called splitting schemes) are constructed for evolutionary equations with general operators. These schemes are based on the extraction of the self-adjoint and skew-symmetric components of the corresponding operator. 相似文献
8.
P. N. Vabishchevich P. E. Zakharov 《Computational Mathematics and Mathematical Physics》2016,56(4):576-592
Explicit–implicit approximations are used to approximate nonstationary convection–diffusion equations in time. In unconditionally stable two-level schemes, diffusion is taken from the upper time level, while convection, from the lower layer. In the case of three time levels, the resulting explicit–implicit schemes are second-order accurate in time. Explicit alternating triangular (asymmetric) schemes are used for parabolic problems with a self-adjoint elliptic operator. These schemes are unconditionally stable, but conditionally convergent. Three-level modifications of alternating triangular schemes with better approximating properties were proposed earlier. In this work, two- and three-level alternating triangular schemes for solving boundary value problems for nonstationary convection–diffusion equations are constructed. Numerical results are presented for a two-dimensional test problem on triangular meshes, such as Delaunay triangulations and Voronoi diagrams. 相似文献
9.
《Journal of Computational and Applied Mathematics》2006,187(2):142-170
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. 相似文献
10.
R. Z. Dautov E. M. Fedotov 《Computational Mathematics and Mathematical Physics》2013,53(11):1614-1625
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. 相似文献
11.
C.V. Pao 《Numerische Mathematik》1995,72(2):239-262
Summary.
Two block monotone iterative schemes for a nonlinear
algebraic system, which is a finite difference approximation of a
nonlinear elliptic boundary-value problem, are presented and are
shown to converge monotonically either from above or from below to
a solution of the system. This monotone convergence result yields
a computational algorithm for numerical solutions as well as an
existence-comparison theorem of the system, including a sufficient
condition for the uniqueness of the solution. An advantage of the
block iterative schemes is that the Thomas algorithm can be used to
compute numerical solutions of the sequence of iterations in the
same fashion as for one-dimensional problems. The block iterative
schemes are compared with the point monotone iterative schemes of
Picard, Jacobi and Gauss-Seidel, and various theoretical comparison
results among these monotone iterative schemes are given. These
comparison results demonstrate that the sequence of iterations from
the block iterative schemes converges faster than the corresponding
sequence given by the point iterative schemes. Application of the
iterative schemes is given to a logistic model problem in ecology
and numerical ressults for a test problem with known analytical
solution are given.
Received
August 1, 1993 / Revised version received November 7, 1994 相似文献
12.
Sadok Lamine Michael G. Edwards 《Journal of Computational and Applied Mathematics》2010,234(7):2106-57
Standard reservoir simulation schemes employ first order upwind schemes for approximation of the convective fluxes when multiple phases or components are present. These convective flux schemes rely upon upwind information that is determined according to grid geometry. As a consequence directional diffusion is introduced into the solution that is grid dependent. The effect can be particularly important for cases where the flow is across grid coordinate lines and is known as cross-wind diffusion.Truly higher dimensional upwind schemes that minimize cross-wind diffusion are presented for convective flow approximation on quadrilateral unstructured grids. The schemes are locally conservative and yield improved results that are essentially free of spurious oscillations. The higher dimensional schemes are coupled with full tensor Darcy flux approximations.The benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full tensor permeability fields. The test cases involve a range of structured and unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce the numerical cross-wind diffusion effect, leading to improved resolution of concentration and saturation fronts. 相似文献
13.
P. N. Vabishchevich 《Computational Mathematics and Mathematical Physics》2013,53(8):1139-1152
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. 相似文献
14.
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. 相似文献
15.
Rooholah Abedian Mehdi Dehghan 《Numerical Methods for Partial Differential Equations》2021,37(1):594-613
In this research, a class of radial basis functions (RBFs) ENO/WENO schemes with a Lax–Wendroff time discretization procedure, named as RENO/RWENO‐LW, for solving Hamilton–Jacobi (H–J) equations is designed. Particularly the multi‐quadratic RBFs are used. These schemes enhance the local accuracy and convergence by locally optimizing the shape parameters. Comparing with the original WENO with Lax–Wendroff time discretization schemes of Qiu for HJ equations, the new schemes provide more accurate reconstructions and sharper solution profiles near strong discontinuous derivative. Also, the RENO/RWENO‐LW schemes are easy to implement in the existing original ENO/WENO code. Extensive numerical experiments are considered to verify the capability of the new schemes. 相似文献
16.
P. N. Vabishchevich 《Computational Mathematics and Mathematical Physics》2010,50(1):112-123
In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are
obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to
time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods
are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the
finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements,
extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues
concerning the computational implementation of schemes having higher approximation orders are discussed. From the general
point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed. 相似文献
17.
一类交错网格的Gauss型格式 总被引:1,自引:0,他引:1
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 . 相似文献
18.
A. B. Al’shin E. A. Al’shina A. G. Limonov 《Computational Mathematics and Mathematical Physics》2009,49(2):261-278
New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff systems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of L1-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single L2-stable scheme is found in this family. The coefficients of the fourth-order accurate L4-stable scheme previously obtained by P.D Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of L-decay. The schemes proposed are proved to converge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions. 相似文献
19.
G. I. Shishkin 《Computational Mathematics and Mathematical Physics》2006,46(11):1953-1971
In an unbounded (with respect to x and t) domain (and in domains that can be arbitrarily large), an initial-boundary value problem for singularly perturbed parabolic reaction-diffusion equations with the perturbation parameter ε2 multiplying the higher order derivative is considered. The parameter ε takes arbitrary values in the half-open interval (0, 1]. To solve this problem, difference schemes on grids with an infinite number of nodes (formal difference schemes) are constructed that converge ε-uniformly in the entire unbounded domain. To construct these schemes, the classical grid approximations of the problem on the grids that are refined in the boundary layer are used. Schemes on grids with a finite number of nodes (constructive difference schemes) are also constructed for the problem under examination. These schemes converge for fixed values of ε in the prescribed bounded subdomains that can expand as the number of grid points increases. As ε → 0, the accuracy of the solution provided by such schemes generally deteriorates and the size of the subdomains decreases. Using the condensing grid method, constructive difference schemes that converge ε-uniformly are constructed. In these schemes, the approximation accuracy and the size of the prescribed subdomains (where the schemes are convergent) are independent of ε and the subdomains may expand as the number of nodes in the underlying grids increases. 相似文献
20.
Monotone absolutely stable conservative difference schemes intended for solving quasilinear multidimensional hyperbolic equations are described. For sufficiently smooth solutions, the schemes are fourth-order accurate in each spatial direction and can be used in a wide range of local Courant numbers. The order of accuracy in time varies from the third for the smooth parts of the solution to the first near discontinuities. This is achieved by choosing special weighting coefficients that depend locally on the solution. The presented schemes are numerically efficient thanks to the simple two-diagonal (or block two-diagonal) structure of the matrix to be inverted. First the schemes are applied to system of nonlinear multidimensional conservation laws. The choice of optimal weighting coefficients for the schemes of variable order of accuracy in time and flux splitting is discussed in detail. The capabilities of the schemes are demonstrated by computing well-known two-dimensional Riemann problems for gasdynamic equations with a complex shock wave structure. 相似文献