In this work, a method for the numerical simulation of free surface flows of Newtonian and non-Newtonian flows is developed. The method is based on the employ of three main ingredients. The first is an updated Lagrangian approach for the description of kinematics. This approach is made possible by using natural neighbor Galerkin methods (also known as natural element methods).This allows the use of the method of characteristics to integrate the equations of motion along the nodal path-lines. The second ingredient relies on the use of a second-order time method of characteristics, which has proved to be indispensable for an accurate solution of some problems, even very simple ones. Finally, the third ingredient relies on the use of shape constructors (particularly, α-shapes) to avoid the use of boundary markers or any explicit description of the boundary in general. After the theoretical description of the proposed method, some examples illustrating its performance are given. Copyright © 2011 John Wiley & Sons, Ltd.