FEUDX: a two-stage, high-accuracy, finite-element FORTRAN program for solving shallow-water equations

Abstract

Abstract A FORTRAN computer program is presented and documented which implements a new, two-stage finite-element Numerov-Galerkin method for integrating the nonlinear shallow-water equations on a β-plane limited-area domain. In this method high accuracy is obtained by combining the Galerkin product with a high-order compact (hence the name Numerov) difference approximation to derivatives in the nonlinear advection operator. Conservation of integral invariants is obtained by nonlinear constrained optimization using the Augmented-Lagrangian method, allowing perfect conservation of the integral invariants for long-term integrations. Program options include the use of a weighted selective lumping scheme in the finite-element method, use of either a Gauss-Seidel or a successive overrelaxation (S.O.R.) iterative method for solving the resulting systems of linear equations, a line-printer plot of the fields contours and finally, determination at each time-step of the values of three integral invariants of the shallow-water equations. A solver for periodic pentadiagonal matrices resulting from the application of the high-order difference approximation is included. Long-term numerical integrations (10–20 days) have been performed using this program. Smallscale noise was eliminated using a Shuman filter, periodically applied to one component of the velocity field. The method was determined to exhibit a consistently higher accuracy than the single-stage finite-element method and can be use to advantage by meteorologists and oceanographers. Due to the code being modular and flexible it can be changed easily to suit the aims of different researchers. A vectorized version of the code, operative on a CYBER-205 also is available.