Numerical modelling of the mild slope equation using localised differential quadrature method

Abstract

Although various numerical techniques have been applied over the last few decades to solve the mild slope equation (MSE), each technique has its own limitations, particularly in terms of computational cost, accuracy, and stability. Localised differential quadrature method (LDQM) is here investigated as an alternative new solution to the MSE. Localised DQM, rather than classical DQM, was used to solve the MSE because of its improved performance, lower computational cost and wider range of applicability. To evaluate the proposed method, four examples were studied, covering a range of complexity which included propagation and transformation of waves due to an elliptic shoal, breakwater gap, and non-rectangular harbour resonance. The results were compared with experimental data, analytical solutions, and other numerical methods. The agreement between numerical and benchmark results was good, and in some cases the performance of LDQM exceeded that of other numerical methods. LDQM can lead to accurate results using fewer grid points and lower computational cost if the number of local nodes is optimised. For a large number of local grid points in LDQM, and also for the case of classical DQM, iterative methods such as conjugate gradient should be employed to solve the system of equations.