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Investigation of Higher-Order Accuracy for a Conservative Semi-Lagrangian Discretization of the Atmospheric Dynamical Equations

Matthew Norman
December 7th, 2007

This study considers higher-order spatial and temporal methods for a conservative semi-implicit semi-Lagrangian (SISL) discretization of the atmospheric dynamical equations. With regard to spatial accuracy, new subgrid approximations are tested in the Conservative Cascade Scheme (CCS) SL transport algorithm. When developed, the CCS used the monotonic Piecewise Parabolic Method (PPM) to reconstruct cell variation. This study adapts four new non-polynomial methods to the CCS context: the Piecewise Hyperbolic Method (PHM), Piecewise Double Hyperbolic Method (PDHM), Piecewise Double Logarighmic Method (PDLM), and Piecewise Rational Method (PRM) for comparison against PPM. Additionally, an adaptive hybrid approximation scheme, PPM-Hybrid (PPM-H), is constructed using monotonic PPM for smooth data and local extrema and using PHM for steep jumps where PPM typically suffers large accuracy degradation. Smooth and non-smooth data profiles are transported in 1-D, 2-D Cartesian, and 2-D spherical frameworks under uniform advection, solid body rotation, and deformational flow. Accuracy is compared in the L_{1} error measure. PHM performed up to five times better than PPM for smooth functions but up to two times worse for non-smooth functions. PRM performed very similarly to PPM for non-smooth functions but the order of convergence was worse than PPM for smooth data. PDHM performed the worst of all of the non-polynomial methods for almost every test case. PPM-H outperformed both PPM and all of the new methods for all test cases in all geometries offering a robust advantage in the CCS scheme. Additionally, the CCS and new subgrid approximations were used to perform conservative grid-to-grid interpolation between two spherical grids in latitude / longitude coordinates. The methods were tested by prescribing an analytical sine wave function which was integrated over grid cells at T-42 resolution (approximately 2.8^{o} imes2.8^{o}) and at 1^{o} resolution. Then, the 1^{o} data is interpolated to the T-42 grid to compare against the analytical formulation. Three test data sets were created with increasing sharpness in the sine wave profiles by spanning 1, 3, and 9 wavelengths across the domain. It was found that in all test cases, PDHM performed the best in the interpolation scheme, better than PPM. Regarding temporal accuracy, a linear, SISL 2-D dynamical model is given harmonic input for the dependent variables to extract a Von-Neumann analysis of the SISL numerical modification of the solution. The Boussinesq approximation is relaxed, and spatial error is removed in order to isolate only temporal accuracy. A hydrostatic switch is employed to invoke and remove non-hydrostatic dynamics. Trajectory uncentering (typically used to suppress spurious orographic SISL resonance) is included by altering the coefficients of the forcing terms of the linear equations. It was found that with regard to Internal Gravity Wave (IGW) motion, the first-, second-, and third-order Adams-Moulton (AM) schemes performed with increasingly greater accuracy. Also, the higher the order of temporal convergence, the greater the gain in accuracy by simulating in a non-hydrostatic context relative to a hydrostatic one. Second-order uncentering resolves IGW phases poorly resulting in an RMSE error nearly the same as the first-order scheme. The third-order AM scheme demonstrated superior accuracy to the other methods in this part of the study. Further research may determine if uncentering is necessary with this method for stability.

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