A note on the application of a time exponential integrator to fully nonlinear numerical wave tanks for engineering applications

This work examines pseudospectral method-based theoretical frameworks for developing Numerical Wave Tanks (NWTs) which predict the spatiotemporal evolution of surface gravity–capillary waves. A special focus is on the numerical implementation. The frameworks are mathematically formulated by generalized and coupled parametric Partial Differential Equations (PDEs). Based on the PDEs, the matrix exponential are newly derived, permitting to explore the method of exponential integrator in developing NWTs using the theoretical frameworks which are reviewed in this work. An envelope-based theoretical framework is particularly newly derived, accurate to the third order in wave steepness. The resulting matrix exponential is validated through applying it in two different theoretical frameworks which are accurate to the first and second order in wave steepness. The exponential integrator method in predicting the phase-resolved evolving waves has a semi-analytical feature since it admits an analytical expression for linear waves propagation and developing approximate approaches for the evolution of nonlinear waves with quantified numerical errors. Hence, it permits a significantly improved numerical accuracy and thereby a wide applicability in engineering applications and physical systems with similar features.