Author(s): Jonatan Mulet Marti; Francisco Alcrudo
Linked Author(s): Jonatan Mulet Martí
Keywords: Finite volume; Shallow Water Equations; Implicit; Riemann solvers; Factorization
Abstract: This work reports the development of implicit algorithms based on the Godunov approach within the finite volume framework to integrate the bidimensional Shallow Water Equations (SWE) in unstructured meshes, and their application to unsteady problems in estuarine environments. There is presently a wealth of modern numerical schemes to accurately solve the SWE which can successfully cope with previously intractable problems such as discontinuities or fronts and extremely abrupt terrain. However, the vast majority are based on explicit time integration schemes which, in order to keep numerical stability, are subject to a time step restriction based on a CFL type condition. Unfortunately, the resulting time steps can be several orders of magnitude smaller than the physical time scales of the phenomenon under consideration and this provides the required level of temporal accuracy very inefficiently. Under these circumstances, the unsteady problem must be solved relentlessly by a series of extremely small time steps to guarantee the numerical stability of the calculation. Implicit time integration methods seem therefore a desirable option, given that the time step restriction can be relaxed and determined mainly by flow physics and accuracy requirements. Implicit schemes have been widely used and are generally the choice in the case of steady state problems, although in the case of the SWE their use has been principally restricted to models based on structured grids. Nevertheless, few references can be found of implicit bidimensional SWE models that solve unsteady problems in unstructured meshes. In this work a novel family of implicit schemes based upon linear multi-step and implicit Runge-Kutta time integration methods is presented, which can be successfully cast in generic polygonal unstructured meshes. The solution procedure relies on a Newton inner iteration whose purpose is to recover the non-linearity of the SWE system every physical time step. An approximate factorization approach has been adopted to overcome the difficulty associated with the solution of large algebraic systems with complex structure. Finally, the application of the method in estuarine environments is presented, analyzed and the results compared against typical explicit schemes.
Year: 2014