The processing technique, while efficient in reducing computational effort and improving accuracy in numerical integration, has some potential limitations compared to other approaches for constructing high-order integrators:
Computational Complexity: As the order of the method increases, the number of effective order conditions to be satisfied also increases. This can lead to more complex calculations and potentially higher computational costs.
Parameter Sensitivity: The optimization process for determining the coefficients of the kernel and processor can be sensitive to the choice of parameters and initial conditions. Small variations in these parameters can impact the efficiency and accuracy of the method.
Limited Applicability: The processing technique is most effective for problems with separable vector fields, where the system can be split into explicitly solvable parts. For more complex systems that do not have this separability property, the processing technique may not be as straightforward to apply.
Processor Approximation: The approximation of the processor may introduce additional errors, especially at higher orders. Ensuring that the processor approximation accurately captures the dynamics of the system can be challenging.