Runge-Kutta type methods for orthogonal integration

A simple characterisation exists for the class of real-valued, autonomous, matrix ODEs where an orthogonal initial condition implies orthogonality of the solution for all time. Here we present first and second order numerical methods for which the property of orthogonality-preservation is always carried through to the discrete approximation. To our knowledge, these are the first methods that guarantee to preserve orthogonality, without the use of projection, whenever it is preserved by the flow. The methods are based on Gauss-Legendre Runge-Kutta formulas, which are known to preserve orthogonality on a restricted problem class. In addition, the new methods are linearly-implicit, requiring only the solution of one or two linear matrix systems (of the same dimension as the solution matrix) per step. Illustrative numerical tests are reported.