scholarly journals A Second-Order Asymptotic-Preserving and Positivity-Preserving Exponential Runge--Kutta Method for a Class of Stiff Kinetic Equations

2019 ◽  
Vol 17 (4) ◽  
pp. 1123-1146 ◽  
Author(s):  
Jingwei Hu ◽  
Ruiwen Shu
Keyword(s):  
Kinetic Equations ◽  
Second Order ◽  
Kutta Method ◽  
Positivity Preserving ◽  
Asymptotic Preserving ◽  
Runge Kutta Method ◽  
Runge Kutta

Symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields

2016 ◽  
Vol 07 (02) ◽  
pp. 1650008 ◽  
Author(s):  
Beibei Zhu ◽  
Zhenxuan Hu ◽  
Yifa Tang ◽  
Ruili Zhang
Keyword(s):  
Numerical Simulation ◽  
Hamiltonian System ◽  
Second Order ◽  
Symplectic Methods ◽  
Kutta Method ◽  
Numerical Accuracy ◽  
Runge Kutta Method ◽  
Simulation Results ◽  
Runge Kutta

We apply a second-order symmetric Runge–Kutta method and a second-order symplectic Runge–Kutta method directly to the gyrocenter dynamics which can be expressed as a noncanonical Hamiltonian system. The numerical simulation results show the overwhelming superiorities of the two methods over a higher order nonsymmetric nonsymplectic Runge–Kutta method in long-term numerical accuracy and near energy conservation. Furthermore, they are much faster than the midpoint rule applied to the canonicalized system to reach given precision.

A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr–Debye model

2017 ◽  
Vol 27 (03) ◽  
pp. 549-579 ◽  
Author(s):  
Juntao Huang ◽  
Chi-Wang Shu
Keyword(s):  
Discontinuous Galerkin ◽  
Strong Stability ◽  
Debye Model ◽  
Second Order ◽  
Positivity Preserving ◽  
Asymptotic Preserving ◽  
Runge Kutta ◽  
Numer Anal ◽  
Scalar Conservation ◽  
Unconditional Positivity

In this paper, we develop a second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) scheme for the Kerr–Debye model. By using the approach first introduced by Zhang and Shu in [Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004) 641–666.] with an energy estimate and Taylor expansion, the asymptotic-preserving property of the semi-discrete DG methods is proved rigorously. In addition, we propose a class of unconditional positivity-preserving implicit–explicit (IMEX) Runge–Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr–Debye model. The new IMEX Runge–Kutta methods are based on the modification of the strong-stability-preserving (SSP) implicit Runge–Kutta method and have second-order accuracy. The numerical results validate our analysis.

An efficient implicit Runge–Kutta method for second order systems

2006 ◽  
Vol 178 (2) ◽  
pp. 229-238 ◽  
Author(s):  
Basem S. Attili ◽  
Khalid Furati ◽  
Muhammed I. Syam
Keyword(s):  
Second Order ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Second Order Systems
Sign in / Sign up

Export Citation Format

Share Document