  In this paper, we establish the higher order convergence rates in periodic
homogenization of viscous Hamilton-Jacobi equations, which is convex and grows
quadratically in the gradient variable. We observe that although the nonlinear
structure governs the first order approximation, the nonlinear effect is
absorbed as an external source term of a linear equation in the second and
higher order approximation. Moreover, we find that the geometric shape of the
initial data has to be chosen carefully according to the effective Hamiltonian,
in order to achieve the higher order convergence rates.
