A recursive system-free single-step temporal discretization method for finite difference methods

Single-stage or single-step high-order temporal discretizations of partial differential equations (PDEs) have shown great promise in delivering accuracy time with efficient use computational resources. There has been much success developing such methods for finite volume method (FVM) PDEs. The Picard Integral formulation (PIF) recently made single-stage accessible difference (FDM) discretizations. PIF rely on the so-called Lax-Wendroff procedures to tightly couple spatial and derivatives through governing PDE system construct Taylor series expansions time. Going higher than third order requires calculation Jacobian-like derivative tensor-vector contractions an increasingly larger degree, greatly adding complexity schemes. To that end, we present this paper a calculating these tensor recursive application discrete Jacobian operator readily efficiently computes needed entirely agnostic being solved.