Cauchy Solution of the Kinematic Wave Shallow-Water Equations Using Square-Grid Finite-Element Method

The kinematic wave system of shallow-water equations is posed as an initial value problem, or Cauchy problem, to investigate error analysis using the square-grid finite-element method. The consistent, lumped, and upwind models of the square finite-element method are analyzed using the Fourier (or von Neumann) stability method as separate initial value problems for two-dimensional shallow-water equations. The small difference between the Cauchy solution and integer (or half) multiples of the eigenvalue solution of amplification factors using the eigenvalue method indicates that the Cauchy solution is a possible stable numerical solution of the kinematic wave shallow-water equations within the reasonable limits of solution accuracy. The nodal amplification factors are less than or equal to unity for implicit finite-element schemes for all of the three formulations—consistent, lumped, and upwind for all of the wave numbers, implying unconditional stability. For explicit and semi-implicit finite-element schemes, at least one of the four nodal amplification factors exceeded unity for all wave numbers, thus implying that the explicit and semi-implicit schemes are unconditionally unstable (except for semi-implicit, upwind). Although element-level amplification factors may be useful in obtaining stable solutions for the interior part of the solution domain, amplification factors exceeding 1.0 might be encountered with associated errors and oscillations. Therefore, an understanding of nodal amplification factors is necessary for better solution accuracy and solution stability. The current research bridges the knowledge gap between the Cauchy and eigenvalue solutions, brings closer the accurate large-scale numerical modeling of watersheds, and delineates directions to be followed in the future for the hydrologic modeling community.