In this work, we apply high order multi-resolution analysis to dete-rmine the smoothness with high order of a function on an equidi-stant grid for the Hybrid scheme.

The adaptive nonlinear reconstruction procedure of WENO schemes avoids interpolations across nonsmooth data, essentially localizing and minimizing numerical oscillations near discontinuities while resolving fine scale structures efficiently in the numerical solution.

We investigate the spectral properties (dispersion and dissipation) of the fifth order nonlinear schemes (the WENO-JS and WENO-Z schemes) using both the approximate dispersion relation and nonlinear spectral analysis.

We investigate the performance of a hybrid 6th-order compact finite difference scheme and 5th-order characteristic-wise weighted essentially non-oscillatory (WENO) finite difference scheme (Hybrid) for the numerical simulations of the detonation waves on a uniformly discretized Cartesian domain.

The Hybrid scheme conjugates a shock-capturing WENO-Z5 non-linear scheme in non-smooth stencils with an essentially non-dissipative and non-dispersive Fourier-Continuation (FC) linear scheme in smooth stencils, for applications containing both discontinuous and complex smooth structures.