Finite-Element Schemes for Extended Integrations of Atmospheric Models

The effect of conservation of integral invariants by finite-element discretization schemes of the shallow-water equations as a model for long-term integrations of atmospheric models is investigated. Two finite-element models are used. The first uses rectangular elements and conserves total energy using an intrinsic method. The second model uses triangular elements and a high-accuracy two-stage Numerov-Galerkin method. It well conserves total energy and potential enstrophy by applying a periodical Shuman filter every 48 or 96 time steps. Different conserving and non-conserving versions of these finite-element schemes are compared in terms of their conservation of the integral invariants of the shallow-water equations for long-term integrations (20100 model days). Critical times for numerical non-linear instability are investigated along with the determination of the critical amount of dissipation required to achieve stable long-term integrations. Comparisons of the two finite-element schemes, namely, the rectangular and the triangular in terms of their relative computational efliciency and accuracy, are also provided. Similarities and differences in the behavior of finite-element schemes and finite-difference schemes for long-term stable integrations of the shallow-water equations are finally discussed. ic“ 1990 Academic Press, Inc