Numerical solution of the Kardar-Parisi-Zhang equation in one, two and three dimensions

On a course-grained level a family of microscopic growth processes may be described by a stochastic differential equation, which is solved numerically for surface dimensions d = 1, 2 and 3. Dimensional analysis shows that the spatial discretization parameter has the meaning of an effective coupling constant. The numerical stability of the Euler integration scheme is discussed. For the strong coupling exponents β defined by surface width ∼ timeβ the following effective values were obtained: β(d = 1) = 0.330 ± 0.004 and β(d = 2) = 0.24 ± 0.005. Considering the width and its ensemble fluctuations at constant dimensionless time the transition between strong and weak coupling phases is located in d = 3. For the largest coupling for which reliable data are available we obtain an effective exponent β close to the best estimates on discrete models, β(d = 3) ∼ 0.17.