Numerical methods for Hermitian unitary differential systems
The paper deals with the numerical integration of unitary differential systems of the form \[ Y'(t)=A(t,Y(t))Y(t), \quad Y(t_0)=Y_0, \quad t\geq t_0, \tag{1} \] where \(Y_0\) is a unitary matrix and \(A(t,Y)\) is skew-Hermitian. The purpose of this paper is to study unitary differential systems whose solutions are also Hermitian for all \(t\) and unitary numerical integrators which are also Hermitian integrators. It is known that in the case when the unitary solution of (1) is also Hermitian for all \(t,\) the known unitary schemes are not generally Hermitian-preserving integrators. The authors examine the known unitary-preserving methods (the \(s\)-stage Gauss-Legendre-Runge-Kutta schemes, projected methods, the Cayley methods, and isospectral methods) to determine whether they are also Hermitian-preserving integrators. It is proved that this is the case for the Cayley unitary-preserving methods. The authors also show how isospectral methods can be used to obtain Hermitian-preserving integrators. The results of numerical tests are provided for some linear and nonlinear problems of Hermitian unitary type for the comparison of different unitary integrators.
\(s\)-stage Gauss-Legendre-Runge-Kutta schemes
scientific article; zbMATH DE number 1398824