The high-order finite difference solver for the incompressible Navier-Stokes equations with free surface dynamics presented in [1] requires solving one Poisson boundary value problems at every Runge-Kutta stage to enforce mass conservation. The corresponding sparse linear systems are typically solved using iterative methods such as GMRES. However, in large-scale simulations, solving such discretized Poisson problem poses a major computational bottleneck, significantly limiting the efficiency and scalability of the overall numerical scheme. In this work, we address this bottleneck by extending the subspace acceleration framework from [2] to improve the initial guess provided to the iterative solver in every time step. As a result, we reach the desired solver tolerance with a lower iteration count. This is achieved by constructing a basis for a low-dimensional subspace of the history matrix from previous solu- tions, utilizing techniques from randomized linear algebra. With high probability, the generated basis is well-conditioned and captures the dominant features of the solution space. Solving the Poisson problem restricted to this subspace in the least-squares sense provides a significantly better initial guess than the common approach of directly reusing the previous solution. Preliminary experiments related to propagation of nonlinear stream function waves show that the proposed subspace acceleration method may reduce the average number of GMRES itera- tions by more than a factor of five compared to using the previous solution as the initial guess. Depending on the problem size and the subspace dimension, this reduction translates into a speed-up of more than a factor of two for solving the Poisson problem. These results highlight the potential of subspace acceleration as a general and robust framework for efficiently solving sequences of large linear systems in time-dependent problems, making it an appealing strategy for computational fluid dynamics applications and beyond. Additional Numerical results from usecases that relates to numerical wave tanks will be pre- sented at the workshop.