Local Piecewise Hyperbolic Reconstruction of Numerical Fluxes for Nonlinear Scalar Conservation Laws

This paper constructs a local third order accurate shock capturing method for hyperbolic scalar conservation laws, based on numerical fluxes with a total variation diminishing (TVD) Runge–Kutta evolution in time, using the idea recently introduced by C. W. Shu and S. J. Osher for essentially nonoscillatory (ENO) methods. The constructed method is an upwind conservative scheme that is local in the sense that numerical fluxes are reconstructed without using extrapolation from the data of the smoothest neighboring cell. To design the method, a new concept of local smoothing is introduced to prevent the increasing of total variation of the solution near discontinuities and to achieve third order accuracy. The method becomes third order accurate in smooth regions of the solution, except at local extrema where it may degenerate to $O(h^{3/2} )$, thus giving better accuracy than TVD methods at local extrema. The main advantage of this method lies on the property that is more local than that of ENO and TVD upwind schemes of the same order, (and, thus, giving better resolution of corners, i.e., jumps in derivative), because numerical fluxes depend only on four variables. Numerical experiments for scalar conservation laws one-dimensional (1D) and two-dimensional (2D) are presented, and they show that the author's method is stable and behaves as an entropy-satisfying method for nonlinear fluxes. This method becomes efficient since it is low cost and it is not sensitive to the Courant–Friedrichs-Lewy (CFL) number.