A novel IMGS-GMRES algorithm for addressing ill-conditioned high-dimensional matrices in hydrodynamic analysis of large floating structures

In large-scale hydrodynamic analyses of large floating structures, conventional iterative algorithms frequently encounter ill-conditioning when solving linear systems derived from boundary element discretisation, typically manifesting as significantly degraded convergence rates or numerical instabilities. To address this challenge, we propose a novel Improved Modified Gram-Schmidt GMRES (IMGS-GMRES) algorithm. This algorithm significantly improves the efficiency and numerical stability in solving ill-conditioned systems by optimising the orthogonalisation process and incorporating preconditioning techniques. The method combines preconditioning strategies with an enhanced Gram-Schmidt orthogonalisation technique to effectively mitigate high condition numbers. The IMGS-GMRES algorithm has been implemented within the in-house hydrodynamic solver FHLAN and validated against the commercial software WADAM, demonstrating improvements in computational accuracy and stability. Furthermore, it optimises the solution process for large-scale linear systems encountered in hydrodynamic analysis. Numerical experiments demonstrate superior performance: for a system with a condition number of 1.06×10²², IMGS-GMRES reduces iterations by 44.8 % (from 498 to 275) and suppresses orthogonality error to the machine precision of 4.11×10⁻¹⁶. In a large-scale case with 31,000 panels, it achieves 82 % runtime reduction vs direct solvers. The algorithm provides an efficient, robust solution for ill-conditioned hydrodynamic analyses of large floating bodies and high-dimensional coupled systems, supporting real-time simulation and optimisation.