In this paper, three versions of WENO schemes WENO-JS, WENO-M and WENO-Z are used for one-dimensional detonation wave simulations with fifth order characteristic based spatial flux reconstruction. Numerical schemes for solving the system of hyperbolic conversation laws using the ZND analytical solution as initial condition are presented. Numerical simulations of one-dimensional detonation wave for both stable and unstable cases are performed. In the stable case with overdrive factor $f=1.8$, the temporal histories of peak pressure of the detonation front computed by WENO-JS and WENO-Z reach the theoretical steady state. In comparison, the temporal history of peak pressure computed by the WENO-M scheme fails to reach and oscillates around the theoretical steady state. In the unstable cases with overdrive factors $f=1.6$ and $f=1.3$, the results of all WENO schemes agree well with each other as the resolution, defined as the number of grid points per half-length of reaction zone, increases. Furthermore, for overdrive factor $f=1.6$, the grid convergence study demonstrates that the high order WENO schemes converge faster than other existing lower order schemes such as unsplit scheme, Roe's solver with minmod limiter and Roe's solver with superbee limiter in reaching the predicted peak pressure. For overdrive factor $f=1.3$, the temporal history of peak pressure shows an increasingly chaotic behavior even at high resolution.  In the case of overdrive factor $f=1.1$, in accordance with theoretical studies, an explosion occurs and different WENO schemes leading to this explosion appear at slightly different times.