The Picard Integral Formulation of Weighted Essentially Nonoscillatory Schemes

Abstract

High-order temporal discretizations for hyperbolic conservation laws have historically been formulated as either a method of lines (MOL) or a Lax--Wendroff method. In the MOL viewpoint, the partial differential equation is treated as a large system of ordinary differential equations (ODEs), where an ODE tailored time integrator is applied. In contrast, Lax--Wendroff discretizations immediately convert Taylor series in time to discrete spatial derivatives. In this work, we propose the Picard integral formulation (PIF), which is based on the method of modified fluxes, and is used to derive new Taylor and Runge--Kutta (RK) methods. In particular, we construct a new class of conservative finite difference methods by applying weighted essentially nonoscillatory (WENO) reconstructions to the so-called time-averaged fluxes. Our schemes are automatically conservative under any modification of the fluxes, which is attributed to the fact that classical WENO reconstructions conserve mass when coupled with forward Euler time steps. The proposed Lax--Wendroff discretization is constructed by taking Taylor series of the flux function as opposed to Taylor series of the conserved variables. The RK discretization differs from classical MOL formulations because we apply WENO reconstructions to time-averaged fluxes rather than taking linear combinations of spatial derivatives of the flux. In both cases, we only need one projection onto the characteristic variables per time step. The PIF is generic and lends itself to a multitude of options for further investigation. At present, we present two canonical examples: one based on Taylor, and the other based on the classical RK method. Stability analyses are presented for each method. The proposed schemes are applied to hyperbolic conservation laws in one and two dimensions and the results are in good agreement with current state-of-the-art methods.