  We study front speeds of curvature and strain G-equations arising in
turbulent combustion. These G-equations are Hamilton-Jacobi type level set
partial differential equations (PDEs) with non-coercive Hamiltonians and
degenerate nonlinear second order diffusion. The Hamiltonian of strain
G-equation is also non-convex. Numerical computation is performed based on
monotone discretization and weighted essentially nonoscillatory (WENO)
approximation of transformed G-equations on a fixed periodic domain. The
advection field in the computation is a two dimensional Hamiltonian flow
consisting of a periodic array of counter-rotating vortices, or cellular flows.
Depending on whether the evolution is predominantly in the hyperbolic or
parabolic regimes, suitable explicit and semi-implicit time stepping methods
are chosen. The turbulent flame speeds are computed as the linear growth rates
of large time solutions. A new nonlinear parabolic PDE is proposed for the
reinitialization of level set functions to prevent piling up of multiple
bundles of level sets on the periodic domain. We found that the turbulent flame
speed $s_T$ of the curvature G-equation is enhanced as the intensity $A$ of
cellular flows increases, at a rate between those of the inviscid and viscous
G-equations. The $s_T$ of the strain G-equation increases in small $A$,
decreases in larger $A$, then drops down to zero at a large enough but finite
value $A_{*}$. The flame front ceases to propagate at this critical intensity
$A_*$, and is quenched by the cellular flow.
