Convergence of Rothe's method for fully nonlinear parabolic equations

Convergence of Rothe’s method for the fully nonlinear parabolic equation u_t + F(D^2 u, Du, u, x, t) = 0 is considered under some continuity assumptions on F. We show that the Rothe solutions are Lipschitz in time, Holder in space, and they solve the equation in the viscosity sense. As an immediate corollary we get Lipschitz behavior in time of the viscosity solutions of our equation.