Runge-kutta Schemes for Hamiltonian Systems
نویسنده
چکیده
We study the application of Runge-Kutta schemes to Hamiltonian systems of ordinary differential equations. We investigate which schemes possess the canonical property of the Hamiltonian flow. We also consider the issue of exact conservation in the time-discretization of the continuous invariants of motion. Classification: AMS 65 L, 70H.
منابع مشابه
Discretization and Weak Invariants
We consider the preservation of weak solution invariants in the time integration of ordinary diier-ential equations (ODEs). Recent research has concentrated on obtaining symplectic discretizations of Hamiltonian systems and schemes that preserve certain rst integrals (i.e. strong invariants). In this article, we examine the connection between constrained systems and ODEs with weak invariants fo...
متن کاملMulti-symplectic Runge–Kutta-type methods for Hamiltonian wave equations
The non-linear wave equation is taken as a model problem for the investigation. Different multisymplectic reformulations of the equation are discussed. Multi-symplectic Runge–Kutta methods and multi-symplectic partitioned Runge–Kutta methods are explored based on these different reformulations. Some popular and efficient multi-symplectic schemes are collected and constructed. Stability analyses...
متن کاملSymplectic and symmetric methods for the numerical solution of some mathematical models of celestial objects
In the last years, the theory of numerical methods for system of non-stiff and stiff ordinary differential equations has reached a certain maturity. So, there are many excellent codes which are based on Runge–Kutta methods, linear multistep methods, Obreshkov methods, hybrid methods or general linear methods. Although these methods have good accuracy and desirable stability properties such as A...
متن کاملGeometric Integrations for Classical Spin Systems
Practical, structure-preserving methods for integrating classical Heisenberg spin systems are discussed. Two new integrators are derived and compared, including (1) a symmetric energy and spin-length preserving integrator based on a Red-Black splitting of the spin sites combined with a staggered timestepping scheme and (2) a (Lie-Poisson) symplectic integrator based on Hamiltonian splitting. Th...
متن کاملExplicit Canonical Methods for Hamiltonian Systems
We consider canonical partitioned Runge-Kutta methods for separable Hamiltonians H = T(ß) + Viq) and canonical Runge-Kutta-Nyström methods for Hamiltonians of the form H = ^pTM~lp + Viq) with M a diagonal matrix. We show that for explicit methods there is great simplification in their structure. Canonical methods of orders one through four are constructed. Numerical experiments indicate the sui...
متن کامل