Finite difference and finite volume ghost multi-resolution WENO schemes with increasingly higher order of accuracy

This article provides the high -order finite difference and finite volume ghost multi -resolution weighted essentially non -oscillatory (GMR-WENO) schemes for solving hyperbolic conservation laws on structured meshes. We only utilize the information defined on one big central spatial stencil without introducing any equivalent multi -resolution representations. These GMR-WENO schemes utilize orthogonal Legendre basis to construct high degree polynomials of big central spatial stencils and use a L2 projection methodology to obtain a series of ghost low degree polynomials whose degree could gradually range from the highest degree to the zeroth degree. Each of these GMR-WENO schemes only utilizes one big central spatial stencil to achieve arbitrary high -order accuracy in smooth regions and could gradually reduce to the first -order accuracy nearby strong discontinuities. The linear weights of such GMR-WENO schemes can be any positive numbers with a summation of one. This is the first time that only one central spatial stencil is utilized in designing high -order finite difference and finite volume WENO schemes. These GMRWENO schemes have a simple construction and can easily achieve arbitrary high -order accuracy in higher dimensions. Benchmark examples are provided to demonstrate the efficiency of these GMR-WENO schemes.