Simulations of Non-Hydrostatic Atmosphere Using Conservation Laws Package

Well-Balanced, Conservative Finite Difference Algorithm for Atmospheric Flows

“…where j is the grid index along the y coordinate (the index along the x coordinate is suppressed for convenience of notation), R L;R G are the reconstruction operators with m and n defining the stencil bounds, and the coefficients for the stencil pointsσ k are functions of the solution-dependent nonlinear weights ω. Equations (23) and 33, representing the WENO5 and CRWENO5 schemes, can be represented through this operator. The subscript G denotes that the nonlinear weights ω are computed based on Gq.…”

“…Although R G represents nonlinear finite difference operators, given by Eqs. (23) and (33), the nonlinearity of these schemes arises from the solution-dependent weights ω k . Within a time-integration step or stage, these weights are computed and fixed, and the operator R G is essentially linear.…”

“…Several algorithms [8,[16][17][18][19][20] solve the governing equations in terms of mass and momentum conservation and use the assumption of adiabaticity to simplify the energy conservation equation to the conservation of potential temperature [21]. Recent efforts solve the Euler equations as conservation laws for mass, momentum, and energy [9,[22][23][24], with no additional assumptions. This form of the governing equations is identical to that used in simulating compressible aerodynamic flows [25,26].…”

Well-Balanced Formulation of Gravitational Source Terms for Conservative Finite-Difference Atmospheric Flow Solvers

“…Alternatively, the assumption of adiabaticity simplifies the energy conservation equation to conservation of the potential temperature; 16 and several numerical algorithms are based on this adiabatic form of the Euler equations. 17,18 Some recent efforts have proposed solving the Euler equations as the conservation of mass, momentum, and energy with no simplifying assumptions, 19,20,9,21 thus allowing the conservation of energy to machine precision, as well as specification of the true viscous terms, should they be required. This approach is followed in the present study.…”

Semi-implicit Time Integration of Atmospheric Flows with Characteristic-Based Flux Partitioning

“…Alternatively, the equations are expressed as the conservation of mass, momentum, and potential temperature [3,27,59,62,68] by assuming adiabatic flows [15]. Recent efforts [2,10,24,28,55] proposed solving the conservation laws for mass, momentum, and energy [41]. If discretized by a conservative numerical method, this approach yields a truly conservative algorithm and allows for the specification of the true viscous terms.…”