A High-Order Accurate, Upwind, and Fully Discrete Linearized Riemann Solver: Application to Non-Hydrostatic Atmospheric Simulation

Matthew Norman, North Carolina State University

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An upwind and fully discrete linearized Riemann solver is proposed for the integration of the compressible non-hydrostatic atmospheric Euler equations (which conserve potential temperature rather than energy). This solver is applicable to any hyperbolic equation set, and we are calling it the Characteristics-Based Flux-Form Semi-Lagrangian (CB-FFSL) method because of its similarities with the FFSL method. Flux-based characteristic variables (a flux-vector splitting analog of Leveque’s f-waves) are reconstructed over upwind stencils for accuracy. The reconstructions are integrated over characteristic trajectories in a Lagrangian fashion to obtain time-averaged fluxes at each interface. Multi-dimensionality is achieved via a standard second-order accurate Strang splitting.

The new solver is evaluated in a finite volume model using a standard set of non-hydrostatic atmospheric test cases. Some advantages of the scheme include low numerical diffusion, minimal parallel communication, computational efficiency, controlled oscillations without filters, and the potential for stability at large Courant numbers. Two non-oscillatory functions are used for reconstructions in the evaluations: a minmod limited linear profile and a third- to fifth-order accurate Weighted Essentially Non-Oscillatory (WENO).

Abstract Author(s): Matthew R. Norman, Ramachandran D. Nair, and Fredrick H. M. Semazzi