Level-set methods applied to the kinematic wave equation governing surface water flows

Abstract

Critical issues arising from the governing nonlinear equations in surface water hydrodynamic include discontinuities in water surface levels, blow-up of water surface gradient, and treatment of dry beds or zero water depths, involving mathematical problems related to functional regularities of unknown variables such as the water depth. The level-set method is a powerful approach to relax requirements for functional regularities of unknowns in nonlinear partial differential equations of first order. In this study, the level-set method is applied to the one-dimensional kinematic wave equation, resulting in a linear level-set equation of the first order in a two-dimensional space to tackle dry beds. The zeros of the level-set function represent the water depths. Hypothesizing that the level-set function is continuous in the domain, it is numerically computed with a characteristic method. The development of overturning is regulated with singular viscosity regularization (SVR), whose effect is to relocate the zeros of the level-set function close to the exact positions of the shock fronts in dam-break problems. The method is firstly verified with the explicitly known exact solutions of primitive dam-break problems, optimizing a parameter of SVR. Then, abrupt water release from Chan Thnal Reservoir, Kampong Speu Province, Cambodia into an initially dry bed of its irrigation canal system is simulated as a practical demonstrative example. In contrast to most of the available software tools using either the shallow water equations with some artificial viscosity or the diffusion wave approximation, the proposed method turns out to be free from spurious diffusive deformation of water surfaces even if relatively coarse computational mesh is used.