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A modified version of explicit Runge-Kutta methods for energy-preserving

Guang-Da Hu (2014)

Kybernetika

In this paper, Runge-Kutta methods are discussed for numerical solutions of conservative systems. For the energy of conservative systems being as close to the initial energy as possible, a modified version of explicit Runge-Kutta methods is presented. The order of the modified Runge-Kutta method is the same as the standard Runge-Kutta method, but it is superior in energy-preserving to the standard one. Comparing the modified Runge-Kutta method with the standard Runge-Kutta method, numerical experiments...

A multilevel method with correction by aggregation for solving discrete elliptic problems

Radim Blaheta (1986)

Aplikace matematiky

The author studies the behaviour of a multi-level method that combines the Jacobi iterations and the correction by aggragation of unknowns. Our considerations are restricted to a simple one-dimensional example, which allows us to employ the technique of the Fourier analysis. Despite of this restriction we are able to demonstrate differences between the behaviour of the algorithm considered and of multigrid methods employing interpolation instead of aggregation.

A multilevel Newton's method for eigenvalue problems

Yunhui He, Yu Li, Hehu Xie, Chun'guang You, Ning Zhang (2018)

Applications of Mathematics

We propose a new type of multilevel method for solving eigenvalue problems based on Newton's method. With the proposed iteration method, solving an eigenvalue problem on the finest finite element space is replaced by solving a small scale eigenvalue problem in a coarse space and a sequence of augmented linear problems, derived by Newton step in the corresponding sequence of finite element spaces. This iteration scheme improves overall efficiency of the finite element method for solving eigenvalue...

A nonlinear system of differential equations with distributed delays

Chocholatý, Pavol (2010)

Programs and Algorithms of Numerical Mathematics

It is well-known that the environments of most natural populations change with time and that such changes induce variation in the growth characteristics of population which is often modelled by delay differential equations, usually with time-varying delay. The purpose of this article is to derive a numerical solution of the delay differential system with continuously distributed delays based on a composition of p -step methods ( p = 1 , 2 , 3 , 4 , 5 ) and quadrature formulas. Some numerical results are presented compared...

A note concerning Gauss-Jackson method.

Ana B. González, Pablo Martín (1996)

Extracta Mathematicae

Specialized literature concerning studies on Orbital Dynamics usually mentions the Gauss-Jackson or sum squared (∑2) method for the numerical integration of second order differential equations. However, as far as we know, no detailed description of this code is available and there is some confusion about the order of the method and its relation with the Störmer method. In this paper we present a simple way of deriving this algorithm and its corresponding analog for first order equations from the...

A numerical scheme for the quantum Boltzmann equation with stiff collision terms

Francis Filbet, Jingwei Hu, Shi Jin (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes was introduced in [F. Filbet and S. Jin,J. Comput. Phys. 229 (2010) 7625–7648] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann equation. To define the quantum Maxwellian...

A numerical scheme for the quantum Boltzmann equation with stiff collision terms⋆

Francis Filbet, Jingwei Hu, Shi Jin (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes was introduced in [F. Filbet and S. Jin,J. Comput. Phys. 229 (2010) 7625–7648] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann...

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