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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...

A ( α )-Stable Linear Multistep Methods for Stiff IVPs in ODEs

R. I. Okuonghae, M. N. O. Ikhile (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In this paper, a class of A( α )-stable linear multistep formulas for stiff initial value problems (IVPs) in ordinary differential equations (ODEs) is developed. The boundary locus of the methods shows that the schemes are A-stable for step number k ≤ 3 and stiffly stable for k = 4 , 5 and 6 . Some numerical results are reported to illustrate the method.

On Runge-Kutta, collocation and discontinuous Galerkin methods: Mutual connections and resulting consequences to the analysis

Vlasák, Miloslav, Roskovec, Filip (2015)

Programs and Algorithms of Numerical Mathematics

Discontinuous Galerkin (DG) methods are starting to be a very popular solver for stiff ODEs. To be able to prove some more subtle properties of DG methods it can be shown that the DG method is equivalent to a specific collocation method which is in turn equivalent to an even more specific implicit Runge-Kutta (RK) method. These equivalences provide us with another interesting view on the DG method and enable us to employ well known techniques developed already for any of these methods. Our aim will...

Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies

Max Duarte, Marc Massot, Stéphane Descombes (2011)

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

In this paper, we investigate the coupling between operator splitting techniques and a time parallelization scheme, the parareal algorithm, as a numerical strategy for the simulation of reaction-diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive...

Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies

Max Duarte, Marc Massot, Stéphane Descombes (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we investigate the coupling between operator splitting techniques and a time parallelization scheme, the parareal algorithm, as a numerical strategy for the simulation of reaction-diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive...

Propagation of errors in dynamic iterative schemes

Zubik-Kowal, Barbara (2017)

Proceedings of Equadiff 14

We consider iterative schemes applied to systems of linear ordinary differential equations and investigate their convergence in terms of magnitudes of the coefficients given in the systems. We address the question of whether the reordering of equations in a given system improves the convergence of an iterative scheme.

The Numerical Solution of Stiff IVPs in ODEs Using Modified Second Derivative BDF

R. I. Okuonghae, M. N. O. Ikhile (2012)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

This paper considers modified second derivative BDF (MSD-BDF) for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The methods are A ( α ) -stable for step length k ≤ 7 .

Time discretizations for evolution problems

Miloslav Vlasák (2017)

Applications of Mathematics

The aim of this work is to give an introductory survey on time discretizations for liner parabolic problems. The theory of stability for stiff ordinary differential equations is explained on this problem and applied to Runge-Kutta and multi-step discretizations. Moreover, a natural connection between Galerkin time discretizations and Runge-Kutta methods together with order reduction phenomenon is discussed.

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