Efficient implementation of high-order isospectral symplectic Runge-Kutta schemes

Abstract: Isospectral Runge-Kutta methods are well-suited for the numerical solution of isospectral systems such as the rigid body and the Toda lattice. More recently, these integrators have been applied to geophysical fluid models, where their isospectral property has provided insights into the long-time behavior of such systems. However, higher-order Isospectral Runge-Kutta methods require solving a large number of implicit equations. This makes the implicit midpoint rule the most commonly used due to its relative simplicity and computational efficiency. In this work, we introduce a novel algorithm that simplifies the implementation of general isospectral Runge-Kutta integrators. Our approach leverages block matrix structures to reduce the number of implicit equations per time step to a single one. This equation can be solved efficiently using fixed-point iteration. We present numerical experiments comparing performance and accuracy of higher-order integrators implemented with our algorithm against the implicit midpoint rule. Results show that, for low-dimensional systems, the higher-order integrators yield improved conservation properties with comparable computational cost. For high-dimensional systems, while our algorithm continues to show better conservation properties, its performance is less competitive, though it can be improved through parallelization.