Numerical methods for non conservative perturbations of conservative problems

Abstract In this paper the numerical integration of non conservative perturbations of differential systems that possess a first integral, as for example slowly dissipative Hamiltonian systems, is considered. Numerical methods that are able to reproduce appropriately the evolution of the first integral are proposed. These algorithms are based on a combination of standard numerical integration methods and certain projection techniques. Some conditions under which known conservative methods reproduce that desirable evolution in the invariant are analysed. Finally, some numerical experiments in which we compare the behaviour of the new proposed methods, the averaged vector field method AVF proposed by Quispel and McLaren and standard RK methods of orders 3 and 5 are presented. The results confirm the theory and show a good qualitative and quantitative performance of the new projection methods.

[1]  M. P. Laburta,et al.  Projection methods preserving Lyapunov functions , 2010 .

[2]  E. Hairer.  Energy-preserving variant of collocation methods 1 , 2010 .

[3]  G. Söderlind,et al.  Geometric integration of Hamiltonian systems perturbed by Rayleigh damping , 2011, 1103.2171.

[4]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[5]  F. Iavernaro,et al.  s‐stage Trapezoidal Methods for the Conservation of Hamiltonian Functions of Polynomial Type , 2007 .

[6]  Brynjulf Owren,et al.  norges teknisk-naturvitenskapelige universitet Preserving multiple first integrals by discrete gradients , 2010 .

[7]  F. Iavernaro,et al.  High-order Symmetric Schemes for the Energy Conservation of Polynomial Hamiltonian Problems 1 2 , 2009 .

[8]  Manuel Calvo,et al.  Energy-preserving methods for Poisson systems , 2012, J. Comput. Appl. Math..

[9]  G. Quispel,et al.  A new class of energy-preserving numerical integration methods , 2008 .

[10]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[11]  Luigi Brugnano,et al.  Line integral methods which preserve all invariants of conservative problems , 2012, J. Comput. Appl. Math..

[12]  Elena Celledoni,et al.  Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method , 2012, J. Comput. Phys..

[13]  Marino Zennaro,et al.  Continuous Explicit Runge-kutta Methods , 1989 .

[14]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[15]  G. Quispel,et al.  Geometric Integration Methods that Preserve Lyapunov Functions , 2005 .

[16]  Oliver Montenbruck,et al.  Satellite Orbits: Models, Methods and Applications , 2000 .

[17]  L. Shampine,et al.  A 3(2) pair of Runge - Kutta formulas , 1989 .

[18]  G. Quispel,et al.  Energy-preserving Runge-Kutta methods , 2009 .

[19]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[20]  Manuel Calvo,et al.  Approximate preservation of quadratic first integrals by explicit Runge–Kutta methods , 2010, Adv. Comput. Math..

[21]  David Cohen,et al.  Linear energy-preserving integrators for Poisson systems , 2011 .

[22]  Manuel Calvo,et al.  Runge–Kutta projection methods with low dispersion and dissipation errors , 2015, Adv. Comput. Math..

[23]  L. Brugnano,et al.  Hamiltonian BVMs (HBVMs): A Family of "Drift Free" Methods for Integrating polynomial Hamiltonian problems' , 2009 .

[24]  G. Quispel,et al.  Geometric integrators for ODEs , 2006 .

[25]  Gábor Geda,et al.  Solving ordinary differential equation systems by approximation in a graphical way , 2006 .

[26]  G. Quispel,et al.  What kinds of dynamics are there? Lie pseudogroups, dynamical systems and geometric integration , 2001 .

[27]  L. Brugnano,et al.  A simple framework for the derivation and analysis of effective one-step methods for ODEs , 2010, Appl. Math. Comput..

[28]  Manuel Calvo,et al.  On the Preservation of Invariants by Explicit Runge-Kutta Methods , 2006, SIAM J. Sci. Comput..