Fractional Governing Equations of Diffusion Wave and Kinematic Wave Open-Channel Flow in Fractional Time-Space. II. Numerical Simulations

Abstract

In this study, finite difference numerical methods, first order accurate in time and second order accurate in space, are proposed to solve the governing equations of the one-dimensional unsteady kinematic and diffusion wave open-channel flow processes in fractional time and fractional space, which were derived in the accompanying paper. Advantages of modeling open-channel flow in a fractional time-space framework over integer time-space framework are threefold. First, the nonlocal phenomena in the open-channel flow process in either space or time can be considered by taking the global correlations into consideration. Second, the proposed governing equations of the open-channel flow process in the fractional order differentiation framework are generalization of the governing equations in the integer order differentiation framework. Third, the physics of the observed heavy tailed distributions of particle displacements in transport processes, as reported in the literature, may be explained by a flow field that is governed by the nonlocal (or long-range dependence) phenomena. Numerical examples in this study demonstrate that the proposed finite difference methods are capable of solving the governing equations of the one-dimensional unsteady kinematic and diffusion wave open-channel flow processes in fractional time and fractional space. The numerical examples also show that the proposed governing equations, which were derived in the accompanying paper for the one-dimensional unsteady kinematic and diffusion wave open-channel flow processes in fractional time and fractional space, may provide additional flexibility and understanding to model open-channel flow processes.