High-order central finite-volume schemes for atmospheric modeling

Atmospheric numerical modeling has been going through drastic changes over the past decade, mainly to utilize the massive computing capability of the petascale systems. This obliges the modelers to develop grid systems and numerical algorithms that facilitate exceptional level of scalability on these systems. The numerical algorithms that can address these challenges should have the local properties such as the high on-processor operation count and minimum parallel communication i.e., high parallel efficiency. They should also satisfy the following properties such as inherent local and global conservation, high-order accuracy, geometric flexibility, non-oscillatory advection and positivity preservation properties. The goal of this dissertation is to address these challenges using various high-order numerical methods. As a possible solution to achieve the above mentioned desirable properties, I considered central-upwind finite-volume (C-FV) schemes, which are proven to be robust, simple and accurate in many research areas and practical applications. These numerical methods are a subset of Godunov-type methods, widely known for their simplicity and for solving hyperbolic conservation laws. But, these staggered central schemes may not provide a satisfactory resolution when small time steps are enforced by stability restrictions. To address this issue, the considered schemes have an upwind nature, in the sense that they are based on the one-sided local speeds of propagation. The central-upwind framework provides high-order accuracy by decreasing the numerical dissipation present in the staggered central schemes. This is the reason, these schemes are central-upwind schemes, but here throughout my present work, I refer them to as C-FV schemes. The construction in the proposed schemes is based on the use of the Courant-Friedrich-Levy (CFL) related local speeds of propagation and on integration over Riemann fans of variable sizes. This way, a non-staggered fully discrete central scheme is derived and is naturally reduced to a particularly simple semi-discrete form. Among the advantages of these schemes are that they do not require Riemann solvers or characteristic decomposition and grid staggering. These characteristics make them different from upwind schemes and universal methods, so they are promising candidate for providing higher-order accuracy to solutions governed by conservative systems. However, little is known for their practicality to geoscience problems that are also governed by conservative systems. In this work, I examine the performance of these high-order schemes. Based on existing knowledge from other fields, I chose five promising schemes that are expected to possess desired properties needed in atmospheric modeling. The five schemes considered are Kurganov-Petrova (KP), a third-order compact central Weighted Essentially Non- Oscillatory (WENO-33), a fifth-order WENO (WENO-5), combination of WENO-33 and WENO-5 (WENO-35), a fourth-order Kurganov-Liu (KL), for a linear transport problem on a two-dimension (2D) Cartesian plane and on sphere. I used the shallow water model on the sphere using the C-FV schemes. The cubed-sphere computational grid system has been chosen in this research work. This type of grid system is very well suited for FV discretization, mainly because the underlying control-volumes (grid cells) are logically rectangular, facilitating an easy implementation. Moreover, the cubed-sphere grid system is free of polar singularities, and its grid uniformity leads to excellent parallel efficiency. For numerical modeling of the transport of trace constituents in atmospheric models, a non-oscillatory positivity preserving solution is essential. Standard WENO schemes produce spurious oscillations in the numerical solution, to address this issue, I employed a Bound-Preserving filter, which restricts the numerical solution to be inside the initial upper and lower bounds and suppresses the spurious oscillations, an additional flux correction step, to achieve strictly positive-definite solution is also employed to remove any negative values produced by the C-FV schemes, I used these filters for KL scheme as well. Both these techniques are inexpensive and effective. I use either a third-order or fourth-order Strong Stability Preserving Runge-Kutta time stepping scheme based on the order of the spatial discretization. The numerical schemes are evaluated with several benchmark tests, on a 2D Cartesian plane and cubed-sphere geometry for transport problem, that accentuate accuracy and conservation properties. In this present work, I only extend three schemes out of five schemes considered for solving the transport equation, i.e. KL, WENO-5 and WENO-35 schemes to shallow water model, because it can be concluded from the results of transport problem on a cubed-sphere geometry that these three schemes perform better than the other schemes in terms accuracy and performance. For evaluating flux for shallow water model, I employ a different flux evaluation formula developed by Kurganov-Noelle-Petrova (KNP), since KNP flux evaluation is more accurate than the one used for transport problem. These three schemes are evaluated using the test suite that is accepted by the atmospheric sciences community