Symplectic numerical integrators in constrained Hamiltonian systems.

*(English)*Zbl 0817.65057This paper concerns a construction of good numerical integrators for a constrained Hamiltonian system of the following form: \(M\dot q= p\), \(\dot p= -\nabla_ q V(q)+ q'(q)^ t\lambda\), \(g(q)= 0\).

The system is differential-algebraic, and moreover it has the symplectic structure. It contains another hidden constraints of the form \(g'(q) M^{-1} p= 0\). The systems of this kind, as for example systems of conservation laws require that the finite difference integrators obey similar rules as the original systems. In the present case, the finite difference method should also have the symplectic structure to give good numerical results.

For the Hamiltonian system defined above two finite difference schemes of the first order with respect to \(p\) and \(q\) are considered. The first one uses the half step for \(p\) and provides the iteration in order to satisfy the constraints \(g(q)= 0\) (it is known as so called SHAKE-type constraints algorithm). This scheme does not satisfy the hidden constraints. The authors prove that this algorithm preserves the wedge product: \(dq_{n+ 1}\wedge dp_{n+ 1}= dq_ n\wedge dp_ n\).

The second scheme (called RATTLE-type constraints algorithm) is a modification of the first one and, by a special hint, assures both constraints to be satisfied. This scheme also preserves the wedge products. A theorem on convergence of both algorithms is proven.

The authors compare these two methods with the nonsymplectic backward differentiation formula (BDF) method for differential-algebraic equations. Conclusion is, that symplectic methods are more efficient than the nonsymplectic BDF method. Both symplectic algorithms give equivalent results.

The system is differential-algebraic, and moreover it has the symplectic structure. It contains another hidden constraints of the form \(g'(q) M^{-1} p= 0\). The systems of this kind, as for example systems of conservation laws require that the finite difference integrators obey similar rules as the original systems. In the present case, the finite difference method should also have the symplectic structure to give good numerical results.

For the Hamiltonian system defined above two finite difference schemes of the first order with respect to \(p\) and \(q\) are considered. The first one uses the half step for \(p\) and provides the iteration in order to satisfy the constraints \(g(q)= 0\) (it is known as so called SHAKE-type constraints algorithm). This scheme does not satisfy the hidden constraints. The authors prove that this algorithm preserves the wedge product: \(dq_{n+ 1}\wedge dp_{n+ 1}= dq_ n\wedge dp_ n\).

The second scheme (called RATTLE-type constraints algorithm) is a modification of the first one and, by a special hint, assures both constraints to be satisfied. This scheme also preserves the wedge products. A theorem on convergence of both algorithms is proven.

The authors compare these two methods with the nonsymplectic backward differentiation formula (BDF) method for differential-algebraic equations. Conclusion is, that symplectic methods are more efficient than the nonsymplectic BDF method. Both symplectic algorithms give equivalent results.

Reviewer: K.Moszyński (Warszawa)

##### MSC:

65L05 | Numerical methods for initial value problems |

70H05 | Hamilton’s equations |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |