High-order spectral method for the simulation of capillary waves with complete order consistency

Abstract We develop the simulation capability for surface capillary waves based on the high-order spectral method, which solves the primitive Euler equations truncated to a prescribed nonlinearity order in wave steepness. The uniqueness in this development is to construct an order-consistent scheme to model the surface tension term, so that the simulation at arbitrary order forms a Hamiltonian system with complete energy conservation. The model equations are integrated in time using an integration factor scheme coupled with a 4th-order Runge-Kutta method (IF-RK4), where the linear terms are solved analytically (i.e., under machine precision in the numerical implementation), and nonlinear terms explicitly. The performance of the model is tested by simulation of a progressive Crapper wave and a broadband capillary wave spectrum. The property of order consistency as well as the enhanced accuracy and stability due to the IF-RK4 (compared to RK4 only) are clearly benchmarked. We finally show the effectiveness of the model to reproduce (for the first time) the thermal equilibrium spectrum of capillary waves when energy is injected at small scales of the wave field.