This paper presents the consistency and stability analyses of the Generalized-a methods applied to nonlinear dynamical systems. The second-order accuracy of this class of algorithms is proved also in the non-linear regime, independently of the quadrature rule for nonlinear internal forces. Conversely, the G-stability notion which is suitable for linear multistep schemes devoted to non-linear dynamic problems cannot be applied, as the non-linear structural dynamics equations are not contractive. Nonetheless, it is proved that the Generalized-a methods are endowed with stability in an energy sense and guarantee energy decay in the high-frequency range as well as asymptotic annihilation. However, overshoot and heavy energy oscillations in the intermediate-frequency range are exhibited. The results of representative numerical simulations performed on relatively simple single-and multiple-degrees-of-freedom non-linear systems are presented in order to confirm the analytical estimates. IntroductionFor many problems in structural dynamics, the time integration of stiff ordinary differential equations is required (Hairer and Wanner 1991, p. 9). Commonly used methods for integrating equations with timescales that differ by several orders of magnitude are implicit as relatively large time steps can be employed. As a matter of fact, most integration schemes are A-stable, i.e. unconditionally stable in the linear regime. Moreover, it is essential that these methods be endowed with mechanisms entailing numerical dissipation in the high-frequency range, with limited algorithmic damping in the low-frequency range. These mechanisms help to eliminate high-frequency modes that are insufficiently resolved by either the spatial discretization, the selected time step or both. Representative members of these algorithms are, among others, the N-b method (Newmark 1959), the HHT-a method (Hilber et al. 1977), the WBZ-a method (Wood et al. 1981), the HP-h 1 method (Hoff and Pahl 1988a, b) and the CH-a method (Chung and Hulbert 1993). These methods exhibit second order accuracy in linear dynamics and permit efficient variable step size techniques, being one-step methods. The CH-a, the HHT-a and the WBZ-a methods, the so-called a-methods, are one-parameter schemes which can be considered as particular cases of a more general class of methods named Generalized-a (G-a) in the foregoing. This class of methods corresponds to the CH-a scheme (Chung and Hulbert 1993), where the algorithmic parameters a m ; a f ; b and c are assumed to be independent of each other.For stiff linear problems, A-stability may not be sufficient to ensure a robust temporal integration. As a matter of fact, some stiff components of the numerical solution damp out very slowly even in the presence of numerical dissipation and can show up oscillations which alter the solution. The low effectiveness of the numerical dissipation and the overshoot consequences on the response of the HHT-a method applied to stiff dynamical systems have been highlighted by Bauchau et al. (1995)...

Abstract. As part of a broader effort to develop nextgeneration models for numerical weather prediction and climate applications, a hydrostatic atmospheric dynamical core is developed as an intermediate step to evaluate a finitedifference discretization of the primitive equations on spherical icosahedral grids. Based on the need for mass-conserving discretizations for multi-resolution modelling as well as scalability and efficiency on massively parallel computing architectures, the dynamical core is built on triangular C-grids using relatively small discretization stencils. This paper presents the formulation and performance of the baseline version of the new dynamical core, focusing on properties of the numerical solutions in the setting of globally uniform resolution. Theoretical analysis reveals that the discrete divergence operator defined on a single triangular cell using the Gauss theorem is only first-order accurate, and introduces grid-scale noise to the discrete model. The noise can be suppressed by fourth-order hyper-diffusion of the horizontal wind field using a time-step and grid-size-dependent diffusion coefficient, at the expense of stronger damping than in the reference spectral model.A series of idealized tests of different complexity are performed. In the deterministic baroclinic wave test, solutions from the new dynamical core show the expected sensitivity to horizontal resolution, and converge to the reference solution at R2B6 (35 km grid spacing). In a dry climate test, the dynamical core correctly reproduces key features of the meridional heat and momentum transport by baroclinic eddies. In the aqua-planet simulations at 140 km resolution, the new model is able to reproduce the same equatorial wave propagation characteristics as in the reference spectral model, including the sensitivity of such characteristics to the meridional sea surface temperature profile.These results suggest that the triangular-C discretization provides a reasonable basis for further development. The main issues that need to be addressed are the grid-scale noise from the divergence operator which requires strong damping, and a phase error of the baroclinic wave at medium and low resolutions.

The properties of C-grid staggered spatial discretizations of the shallow-water equations on regular Delaunay triangulations on the sphere are analyzed. Mass-conserving schemes that also conserve either energy or potential enstrophy are derived, and their features are analogous to those of the C-grid staggered schemes on quadrilateral grids. Results of numerical tests carried out with explicit and semi-implicit time discretizations show that the potential-enstrophy-conserving scheme is able to reproduce correctly the main features of large-scale atmospheric motion and that power spectra for energy and potential enstrophy obtained in long model integrations display a qualitative behavior similar to that predicted by the decaying turbulence theory for the continuous system.

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We present an adaptive discretization approach for model equations typical of numerical weather prediction (NWP), which combines the semi-Lagrangian technique with a semiimplicit time discretization method, based on the Trapezoidal Rule second-order Backward Difference Formula scheme (TR-BDF2), and with a discontinuous Galerkin (DG) spatial discretization, with variable and adaptive element degree. The resulting method has full second-order accuracy in time, can employ polynomial bases of arbitrarily high degree in space, is unconditionally stable and can effectively adapt the number of degrees of freedom employed in each element at runtime, in order to balance accuracy and computational cost. Furthermore, although the proposed method can be implemented on arbitrary unstructured and non-conforming meshes, even its application on simple Cartesian meshes in spherical coordinates can reduce the impact of the coordinate singularity, by reducing the polynomial degree used in the polar elements. Numerical results are presented, obtained on classical benchmarks with two-dimensional models implementing discretizations of the shallowwater equations on the sphere and of the Euler equations on a vertical slice, respectively. The results confirm that the proposed method has a significant potential for NWP applications.