Symplectic Lanczos and Arnoldi Method for Solving Linear Hamiltonian Systems: Preservation of Energy and Other Invariants

Krylov subspace methods have become popular for the numerical approximation of matrix functions as for example for the numerical solution of large and sparse linear systems of ordinary differential equations. One well known technique is based on the method of Arnoldi which computes an orthonormal basis of the Krylov subspace. However, when applied to Hamiltonian linear systems of ODEs, this method fails to preserve the symplecticity of the solution under numerical discretization, or to preserve energy. In this work we apply the Symplectic Lanczos Method to construct a J-orthogonal basis of the Krylov subspace. This basis is then used to construct a numerical approximation which is energy preserving. The symplectic Lanczos method is widely used to approximate eigenvalues of large and sparse Hamiltonian matrices, but the approach for solving linear Hamiltonian systems is not well known in the literature. We also show that under appropriate additional assumptions on the structure of the linear Hamiltonian system, the Arnoldi method can preserve certain invariants of the system. We finally investigate numerically the energy and global error behaviour for the methods.