A new collision operator for lattice Boltzmann shallow water model: a convergence and stability study

•A method solving shallow water equations based on a cumulant collision operator.•Accuracy of the new method comparable to the accuracy of the classical BGK approach.•Overcoming of stability problems of BGK method also for low viscosity values.•Conservation of the correct viscosity and phase with second order convergence.•Allowing simulations of natural phenomena involving the water propagation.