Gustaf: A quasi-newton nonlinear ADI FORTRAN IV program for solving the shallow-water equations with augmented lagrangians

Abstract

A FORTRAN IV computer program is documented which implements the nonlinear alternating direction implicit (ADI) method of Gustafsson (1971) for a limited area finite-difference integration of a shallow-water equations model on a β-plane. In this method a computationally efficient quasi-Newton method is used to solve, at each time-step, the resulting nonlinear systems of algebraic equations. Large time-steps can be employed with this method, which is stable unconditionally for the linearized equations. Owing to its nonlinearity, the method is useful particularly where accuracy is important. An augmented Lagrangian method is applied to enforce conservation of the integral invariants of the shallow-water equations. This method approximates the nonlinearly constrained minimization problem by solving a series of unconstrained minimization problems. Program options include a line-printer plot of the height-field contour and determination, at each time-step, of the three integral invariants of the shallow-water equations. According to the number of nonlinear quasi-Newton (QN) iterations performed at each time-step, different QN methods are presented. Long-term runs have been performed using this program and, due to the enforcement of conservation of integral-invariants via the augmented Lagrangian method, no finite-time “blowing” was experienced.