Computation of steady gravity–capillary waves on deep water based on the pseudo-arclength continuation method

Abstract

Although the pseudo-arclength continuation method is classical, this method has not been widely or effectively applied in the area of the numerical calculation of nonlinear waves in fluids. The author believes that the present study is the first work that provides detailed numerical procedures incorporating the pseudo-arclength continuation method with the computation of steady solutions of nonlinear model equations for both 2-D and 3-D viscous gravity–capillary waves on deep water generated by a moving pressure forcing. The basic mathematical concept of the pseudo-arclength continuation method, which originated from the study of solving a nonlinear ordinary differential equation (ODE), is introduced, and the associated numerical procedure in solving the present partial differential equation (PDE) is subsequently presented. Numerical results are shown in terms of steady response diagrams and associated nonlinear wave solution profiles to understand the interrelated effect of forcing, nonlinearity, and viscous damping. In the steady response diagram, once an initial point is determined, then all of the possible nonlinear wave solutions are found, making a complex branch in the diagram. In particular, if the viscous effect is included, however small, then the branch shows a zig-zag pattern, i.e., several turning points, in which multiple steady wave solution profiles are possible for a certain forcing speed. The proposed method can also be applied to any types of PDEs or ODEs in other areas where nonlinearities cannot be neglected.