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Abstract: The system of shallow water equations admits infinitely many conservation laws. This paper investigates weak local residuals as smoothness indicators of numerical solutions to the shallow water equations. To get a weak formulation, a test function and integration are introduced into the shallow water equations. We use a finite volume method to solve the shallow water equations numerically. Based on our numerical simulations, the weak local residual of a simple conservation law with a simple test function is identified to be the best as a smoothness indicator. [Copyright &y& Elsevier]

In this paper, we will extend the lattice Boltzmann model (LBM) to deal with the complex coupled nonlinear Schrödinger equations. A D1Q5 nonstandard complex lattice Boltzmann model (CLBM) with amending term for the coupled nonlinear Schrödinger equations (CNLSEs) is proposed. Complex equilibrium distribution functions and amending terms are selected, and the CNLSEs can be recovered by using the Chapman–Enskog multi-scale analysis and Taylor expansion technique. Some initial boundary value problems of the coupled system are simulated to verify the effectiveness and stability of our model. [ABSTRACT FROM AUTHOR]