An Implicit Nonlinearly Consistent Method for the Two-Dimensional Shallow-Water Equations with Coriolis Force

Abstract

An implicit and nonlinearly consistent (INC) solution technique is presented for the two-dimensional shallow-water equations. Since the method is implicit, and therefore unconditionally stable, time steps may be used that result in both gravity wave Courant?Friedrichs?Lewy (CFL) numbers and advection CFL numbers being larger than one. By nonlinearly consistent it is meant that all of the unknown variables appear at the same time level in the equations and are solved for simultaneously in an iterative manner (i.e., no splitting errors in time). The INC method is compared to a more traditional semi-implicit method for stepping over the gravity wave stability constraint. Results are presented that show that the second-order-in-time INC method can maintain a high level of accuracy if the dynamical timescale of the system is resolved by the time step. To investigate this difference in temporal integration accuracy between the nonlinearly consistent method and the semi-implicit method an approximate modified equation analysis was performed. The INC solution technique employed is the Jacobian-free Newton?Krylov (JFNK) method. Preconditioning of the JFNK method is achieved via a semi-implicit solution method. The height matrix from the semi-implicit algorithm is solved using a multigrid linear solver to provide efficiency and scalability.