### Citations

710 |
Solving Ordinary Differential Equations I
- Hairer, Nørsett, et al.
- 1987
(Show Context)
Citation Context ...provide an example. Taking into account the specific features of time-dependent problems for PDEs, we are interested in numerical methods for solving the Cauchy problem in the case of stiff equations =-=[5,6,7]-=-. When time-dependent problems are solved approximately, the accuracy can be improved in various ways. In the case of two-level schemes (the solution at two adjacent time levels is involved), polynomi... |

380 |
Numerical initial value problems in ordinary differential equations. Prentice-Hall: Englewood Cliffs
- Gear
- 1971
(Show Context)
Citation Context ...s itself in the approximation of time derivatives with a higher accuracy on a multipoint stencil. A characteristic example is provided by multistep methods based on backward numerical differentiation =-=[9]-=-. Various classes of stable finite difference schemes can be constructed to obtain a numerical solution [10,11]. It is important to select among all stable schemes such a scheme that is optimal in ter... |

292 |
Numerical Methods for Ordinary Differential Equations
- Butcher
- 2003
(Show Context)
Citation Context ...provide an example. Taking into account the specific features of time-dependent problems for PDEs, we are interested in numerical methods for solving the Cauchy problem in the case of stiff equations =-=[5,6,7]-=-. When time-dependent problems are solved approximately, the accuracy can be improved in various ways. In the case of two-level schemes (the solution at two adjacent time levels is involved), polynomi... |

265 |
Numerical Solution of TimeDependent Advection-Diffusion-Reaction Equations
- Hundsdorfer, Verwer
- 2003
(Show Context)
Citation Context ...approximations. 1 Introduction When time-dependent problems of mathematical physics are solved numerically, much emphasis is placed on computational algorithms of higher orders of accuracy (e.g., see =-=[1,2]-=-). Along with improving the approximation accuracy with respect to space, improving the approximation accuracy with respect to time is also of interest. In this respect, the results concerning the num... |

154 |
Numerical Solutions of Convection-Diffusion problems
- Morton
- 1996
(Show Context)
Citation Context ...conservation property of the solution (conservation law), the neutral stability of the solution. 1. Upwind (directional) approximations of first order. To approximate the convective terms (see, e.g., =-=[1,15,16]-=-), the upwind first-order approximations are traditionally widely used. In this case, the grid convection operator C has the form C = ∂−. (13) Operator C defined in (13) is non-negative (C ≥ 0). In th... |

140 |
Introduction to the Theory of Difference Schemes
- Samarskii
- 1971
(Show Context)
Citation Context ...eristic example is provided by multistep methods based on backward numerical differentiation [9]. Various classes of stable finite difference schemes can be constructed to obtain a numerical solution =-=[10,11]-=-. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In the theory of finite difference schemes, there is the class of asymptotic... |

122 |
Stability of Runge-Kutta methods for stiff nonlinear differential equations
- DEKKERAND, VERWER
- 1984
(Show Context)
Citation Context ...ime levels is involved), polynomial approximations of the scheme operators on the solutions are used explicitly or implicitly. The most popular representatives of such schemes are Runge-Kutta methods =-=[7,8]-=-, which are widely used in modern computations. The main feature of the multilevel schemes (multistep methods) manifests itself in the approximation of time derivatives with a higher accuracy on a mul... |

109 |
Finite difference methods for ordinary and partial differntial equations
- Leveque
- 2007
(Show Context)
Citation Context ...spect to space, improving the approximation accuracy with respect to time is also of interest. In this respect, the results concerning the numerical methods for ordinary differential equations (ODEs) =-=[3,4]-=- provide an example. Taking into account the specific features of time-dependent problems for PDEs, we are interested in numerical methods for solving the Cauchy problem in the case of stiff equations... |

49 |
Numerical Computation of Internal and External Flows: Fundamentals of Numerical Discretization
- HIRSCH
- 1988
(Show Context)
Citation Context ...y (fulfillment of the maximum principle). The most interesting attempts in the class of linear approximations are associated with the use of approximations with the upwind differences of second order =-=[2,17]-=-. For our problem (4)–(6), we have Cy = 3yi − 4yi−1 + yi−2 2h . Using previously introduced operator notations we obtain C = ∂− + h 2 ∂−∂−. (17) Operator C ≥ 0 and so for the solution of problem (10),... |

34 | Homogeneous difference schemes
- Tikhonov, Samarskii
(Show Context)
Citation Context ...ct among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In the theory of finite difference schemes, there is the class of asymptotically stable schemes (see =-=[12,13]-=-) that ensure the correct long-time behavior of the approximate solution. In the theory of numerical methods for ODEs (see [7,9]), the concept of L-stability is used, which reflects the long-time asym... |

26 |
High Order Difference Methods for Time Dependent PDE
- Gustafsson
- 2008
(Show Context)
Citation Context ...approximations. 1 Introduction When time-dependent problems of mathematical physics are solved numerically, much emphasis is placed on computational algorithms of higher orders of accuracy (e.g., see =-=[1,2]-=-). Along with improving the approximation accuracy with respect to space, improving the approximation accuracy with respect to time is also of interest. In this respect, the results concerning the num... |

21 |
Numerical Methods for Evolutionary Differential Equations
- Ascher
- 2008
(Show Context)
Citation Context ...spect to space, improving the approximation accuracy with respect to time is also of interest. In this respect, the results concerning the numerical methods for ordinary differential equations (ODEs) =-=[3,4]-=- provide an example. Taking into account the specific features of time-dependent problems for PDEs, we are interested in numerical methods for solving the Cauchy problem in the case of stiff equations... |

2 |
Numerical Methods for Solving the stiff systems
- Rakitsky, Ustinov, et al.
- 1979
(Show Context)
Citation Context ...provide an example. Taking into account the specific features of time-dependent problems for PDEs, we are interested in numerical methods for solving the Cauchy problem in the case of stiff equations =-=[5,6,7]-=-. When time-dependent problems are solved approximately, the accuracy can be improved in various ways. In the case of two-level schemes (the solution at two adjacent time levels is involved), polynomi... |

1 |
Difference Schemes with Operator Factors. Dordrecht Hardbound
- Samarskii, Matus, et al.
- 2002
(Show Context)
Citation Context ...eristic example is provided by multistep methods based on backward numerical differentiation [9]. Various classes of stable finite difference schemes can be constructed to obtain a numerical solution =-=[10,11]-=-. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In the theory of finite difference schemes, there is the class of asymptotic... |

1 |
Two-Level Finite Difference Scheme of Improved Accuracy Order for Time-Dependent Problems
- Vabishchevich
(Show Context)
Citation Context ...e theory of numerical methods for ODEs (see [7,9]), the concept of L-stability is used, which reflects the long-time asymptotic behavior of the approximate solution from a different point of view. In =-=[14]-=- there are considered properties of two-level difference schemes of high order approximation for the approximate solution of the Cauchy problem for evolutionary equations with self-adjoint operators. ... |

1 |
Vabishchevich P. N.Methods for Convection-Diffusion Problems
- Samarskii
- 2004
(Show Context)
Citation Context ...conservation property of the solution (conservation law), the neutral stability of the solution. 1. Upwind (directional) approximations of first order. To approximate the convective terms (see, e.g., =-=[1,15,16]-=-), the upwind first-order approximations are traditionally widely used. In this case, the grid convection operator C has the form C = ∂−. (13) Operator C defined in (13) is non-negative (C ≥ 0). In th... |