The RAIL scheme provides theoretical guarantees for high-order accuracy and stability, particularly through its design that incorporates implicit and implicit-explicit (IMEX) time discretizations. The first-order RAIL scheme has been shown to be consistent with the augmented BUG integrator, which has established local error bounds. The local error bound indicates that the error is proportional to the time step size and the truncation tolerance, ensuring that the scheme remains stable under the conditions of Lipschitz continuity and boundedness of the differential operators.
For higher-order RAIL schemes, while rigorous error bounds are still a topic of ongoing research, numerical experiments have demonstrated high-order accuracy. The inclusion of bases from all previous Runge-Kutta stages in the reduced augmentation process enriches the approximation space, allowing for better representation of the solution dynamics and thus supporting high-order accuracy.
In comparison to other low-rank integrators, such as the projected Runge-Kutta methods, the RAIL scheme offers a more flexible framework that can adapt to both implicit and explicit treatments of stiff and non-stiff terms. Projected Runge-Kutta methods typically focus on explicit time-stepping and may not handle stiffness as effectively as the RAIL scheme, which is designed to accommodate implicit treatments. This adaptability, combined with the potential for high-order accuracy, positions the RAIL scheme as a robust alternative in the landscape of low-rank integrators.