A fifth-order conservative compact weighted essentially non-oscillatory hybrid scheme

The compact finite difference (FD) scheme provides superior accuracy and spectral properties compared to explicit methods using the same stencil. It is well known that linear schemes with accuracy higher than second order cannot handle discontinuities, and compact schemes are no exception. A robust approach to capture shock is to combine compact schemes with the weighted essentially non-oscillatory (WENO) scheme, which is currently the most successful high-accuracy approach for shock-capturing. However, two issues arise in the current combination process. First, integrating the compact scheme with the WENO scheme expands the computational stencil. Second, solving a block-tridiagonal system of equations is required when implementing the compact-WENO hybrid schemes in characteristic space. To address these limitations, this paper proposes a new compact-WENO hybrid scheme based on the framework of the conservative FD method. We first develop a fifth-order linear compact scheme to interpolate cell-node values using cell-center values. Numerical fluxes can then be calculated using a fifth-order explicit scheme, which contains the values of three cell centers and two cell nodes. Although the difference scheme uses five points, its stencil contains only three cells. Furthermore, since the WENO scheme is implemented in the explicit flux calculation, there is no need to solve a block-tridiagonal system of equations during the characteristic-wise reconstruction. This design avoids problems faced by the traditional compact-WENO hybrid schemes. Numerical results validate the computational efficiency, the designed order of accuracy, and the shock-capturing performance of the new scheme.