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

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