The new processed composition methods can be extended to higher orders beyond 6th-order by following a similar approach to the one used for constructing the 4th and 6th-order schemes.
Effective Order Conditions: For each higher order, a set of effective order conditions needs to be derived based on the Lie formalism. These conditions will determine the coefficients of the kernel and the processor.
Kernel Construction: The kernel of the composition method can be built by minimizing an objective function that captures the error terms at higher orders. By exploring the parameter space and identifying local minima, the coefficients of the kernel can be determined.
Processor Approximation: The processor can be approximated by a composition of the same form as the kernel, with the coefficients chosen to satisfy the effective order conditions up to a certain order.
Overall Method: The final integrator is then constructed by combining the kernel and the processor in a processed composition method, ensuring that the overall method achieves the desired order of accuracy.
By following these steps and optimizing the coefficients for higher orders, the new processed composition methods can be extended to achieve higher-order accuracy in numerical integration.