A single-step third-order temporal discretization with Jacobian-free and Hessian-free formulations for finite difference methods

Discrete updates of numerical partial differential equations (PDEs) rely on two branches temporal integration. The first branch is the widely-adopted, traditionally popular approach method-of-lines (MOL) formulation, in which multi-stage Runge-Kutta (RK) methods have shown great success solving ordinary (ODEs) at high-order accuracy. clear separation between and spatial discretizations governing PDEs makes RK highly adaptable. In contrast, second formulation using so-called Lax-Wendroff procedure escalates use tight couplings derivatives to construct approximations advancements Taylor series expansions. last decades, modern explored route extensively proposed a set computationally efficient single-stage, single-step accurate algorithms. this paper, we present an algorithmic extension method called Picard integration (PIF) that belongs updates. presented paper furnishes ease calculating Jacobian Hessian terms necessary for third-order accuracy time.