Improved explicit finite difference method for extended shallow water partial differential equation

The shallow water and Boussinesq equations, being highly nonlinear coupled partial differential equations, form a significant basis for the simulation of flow phenomena within the surface and subsurface domains. The application of these equations extends into several fields in the area of engineering and science, including the simulation of Kelvin wake waves, scenarios involving dam break, propagation of flood waves and flow over a bump. In this paper, the shallow water equation for surface flow regions and the Boussinesq partial differential equation for subsurface flow regions are unified into a single set of extended shallow water nonlinear partial differential equations which is applicable for both flow domains. These equations are solved numerically using an improved two-step Lax-Wendroff method with sixth-order accuracy. The method is validated by benchmarking against existing experimental data, which shows a percentage error of less than 1%, confirming its high accuracy and reliability. Moreover, the method is applied to two test cases. The first case is flow over a bump, where the relative error for surface and subsurface flow regions is as low as 10 −8 at t = 100. In the second case, Kelvin wake wave is investigated with the improved sixth-order Lax-Wendroff scheme predicting an arm angle of 20.6955°, while a theoretical particle angle of 19.47° was observed. The results show that this method is good enough and useful for the simulation of surface and subsurface flows phenomena.