  Many reaction-diffusion systems in various applications exhibit traveling
wave solutions that evolve on multiple spatio-temporal scales. These traveling
wave solutions are crucial for understanding the underlying dynamics of the
system. In this work, we present sixth-order weighted essentially
non-oscillatory (WENO) methods within the finite difference framework to solve
reaction-diffusion systems. The WENO method allows us to use fewer grid points
and larger time steps compared to classical finite difference methods. Our
focus is on solving the reaction-diffusion system for the traveling wave
solution with the sharp front. Although the WENO method is popular for
hyperbolic conservation laws, especially for problems with discontinuity, it
can be adapted for the equations of parabolic type, such as reaction-diffusion
systems, to effectively handle sharp wave fronts. Thus, we employed the WENO
methods specifically developed for equations of parabolic type. We considered
various reaction-diffusion equations, including Fisher's, Zeldovich,
Newell-Whitehead-Segel, bistable equations, and the Lotka-Volterra
competition-diffusion system, all of which yield traveling wave solutions with
sharp wave fronts. Numerical examples in this work demonstrate that the central
WENO method is highly more accurate and efficient than the commonly used finite
difference method. We also provide an analysis related to the numerical speed
of the sharp propagating front in the Newell-Whitehead-Segel equation. The
overall results confirm that the central WENO method is highly efficient and is
recommended for solving reaction-diffusion equations with sharp wave fronts.
