Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I

We approximate quasi-linear parabolic SPDEs substituting the derivatives in the space variable with finite differences. When the nonlinear terms in the equation are Lipschitz continuous we estimate the rate of L-p convergence of the approximations and we also prove their almost sure uniform convergence to the solution. When the nonlinear terms are not Lipschitz continuous we obtain this convergence in probability, if the pathwise uniqueness for the equation holds.