Subspace Acceleration for Efficient Nonlinear Water Wave Simulation

TL;DR

This paper introduces a subspace acceleration technique that enhances initial guesses for iterative solvers, significantly reducing computation time in nonlinear water wave simulations while maintaining accuracy.

Contribution

The authors extend existing subspace acceleration methods by incorporating full solution history with exponential weighting, improving efficiency and scalability in solving the Poisson problem.

Findings

Reduces GMRES iterations in water wave simulations

Achieves computational efficiency without loss of accuracy

Applicable across various discretization methods

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.