Subspace Acceleration for Efficient Nonlinear Water Wave Simulation

TL;DR

This work tackles the computational bottleneck in simulating nonlinear, dispersive free-surface flows by repeatedly solving time-dependent Poisson problems for pressure within incompressible Navier–Stokes solvers. It introduces subspace acceleration that leverages the smooth temporal evolution of successive solves to generate high-quality initial guesses, using either a randomized range finder or an exponentially weighted moving average (EWMA) to construct a low-dimensional subspace from previous solutions. The method yields significant reductions in GMRES iterations per time step across benchmark problems (stream-function waves and a submerged bar), while remaining discretization-agnostic and compatible with high-order finite-difference, method-of-lines, and low-storage Runge–Kutta time integration. The approach promises practical gains in runtime and scalability for large-scale free-surface wave simulations and is poised for implementation in HPC environments and more complex wave–structure interaction studies.

Abstract

Efficient simulation of nonlinear and dispersive free-surface flows governed by the incompressible Navier-Stokes equations remains a central challenge in ocean and coastal engineering. The computational bottleneck arises from solving a time-dependent discretized Poisson problem at every time step to enforce divergence free flow. This is crucial to ensure conservation of mass and requires solving long sequences of time-dependent linear systems typically using iterative methods, such as the preconditioned Krylov subspace methods. In this work, we investigate new subspace acceleration techniques for improving initial guesses to reduce the number of iterations required by iterative solvers, with a focus on nonlinear wave propagation problems. We extend the original subspace acceleration method by incorporating the complete history of previous solutions through an exponentially weighted formulation. This approach eliminates the need for repeated sketching and orthonormalization, resulting in a more efficient and scalable strategy to generate better initial guesses. Our method is implemented within a high-order finite-difference framework using a method-of-lines formulation and a low-storage Runge-Kutta time integration scheme. We demonstrate that subspace acceleration significantly reduces the number of GMRES iterations when solving the Poisson equation in nonlinear water wave simulations. Performance is evaluated on two benchmark problems: nonlinear stream function wave propagation and harmonic wave generation over a submerged bar. In both cases, the new approach achieves substantial improvements in computational efficiency without compromising accuracy. Although demonstrated using high-order finite difference methods, the technique is discretization independent and broadly applicable to incompressible free-surface flow solvers.

Figures (5)

• Figure 1: The domain $\Omega_t$ at time $t$ with free surface $\eta(x,t)$ and still-water depth $h(x)$ in two spatial dimensions.
• Figure 2: The figure shows the average number of GMRES iterations at every time-step to solve the linear system of equations using an ILU-preconditioned GMRES for different wave parameters.
• Figure 3: The figure shows the average time used at every time-step to compute the initial guess and to solve the linear system of equations using an ILU-preconditioned GMRES.
• Figure 4: Longitudinal section of wave flume and the location of gauges used for the harmonic wave generation over a submerged bar.
• Figure 5: The figure shows the simulated flow over the submerged bar and experimental data at the gauges.

Theorems & Definitions (1)

• Theorem 4.1