# The recursive system-free method


## The original system-free approach

Following the same algorithmic strategy, the fourth-order extension of the PIF method requires the third-order derivatives of the flux function. Theoretically speaking, the original SF method can approximate this term as,





## The recursive approach





Note that the above improved version of Hessian-free method is applicable regardless the tensor contraction with two identical vectors or with two distinct vectors, hence it does not require separate formulations needed in the original SF approach.




The recursive SF method successfully extended the PIF method [3] to the fourth-order accuracy [1], and you can find the implementations in the SlugCode. The detailed numerical test results are presented in the published article.

 [1] Lee, Y., Lee, D., & Reyes, A. (2021). A recursive system-free single-step temporal discretization method for finite difference methods. Journal of Computational Physics: X, 12, 100098. https://doi.org/10.1016/j.jcpx.2021.100098
 [2] Lee, Y., & Lee, D. (2021). A single-step third-order temporal discretization with Jacobian-free and Hessian-free formulations for finite difference methods. Journal of Computational Physics, 427, 110063. https://doi.org/10.1016/j.jcp.2020.110063
 [3] (1, 2) Christlieb, A. J., Guclu, Y., & Seal, D. C. (2015). The Picard integral formulation of weighted essentially nonoscillatory schemes. SIAM Journal on Numerical Analysis, 53(4), 1833-1856. https://doi.org/10.1137/140959936