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

Highlights: • A new single-step, third-order temporal method extending the Picard Integration Formulation to a system-free scheme. • The new SF-PIF introduces new Jacobian-free and Hessian-free formulations for flux Jacobian and Hessian tensor calculations. • The new single-step SF-PIF method performs twice faster than the three-stage SSP-RK3 solver. In this paper, we present an algorithmic extension of the method called the Picard Integration Formulation (PIF) that belongs to temporal updates based on the Lax-Wendroff procedure. The new extension is called the system-free (SF) approach, which furnishes ease of calculating the Jacobian and the Hessian terms necessary for third-order temporal accuracy in the original PIF method. In contrast to the analytical calculations of the Jacobian and the Hessian tensor terms in the original PIF method, our new SF approach utilizes finite difference approximations that replace the analytical calculations of the Jacobian and Hessian terms with Jacobian-free and Hessian-free approximations in a way commonly adopted in the context of iterative methods. The resulting SF approach enables our new PIF method to be a computationally efficient single-step, third-order accurate temporal scheme, whose computational performance is twice faster than the three-stage SSP-RK3 method with the same accuracy.