A Scalable Fully Implicit Compressible Euler Solver for Mesoscale Nonhydrostatic Simulation of Atmospheric Flows

Abstract

Abstract. A fully implicit solver is developed for the mesoscale nonhydrostatic simulation of atmospheric flows governed by the compressible Euler equations. To spatially discretize the Euler equations on a height-based terrain-following mesh, we apply a cell-centered finite volume scheme, in which an AUSM + -up method with a piecewise linear reconstruction is employed to achieve secondorder accuracy for the low-Mach flow. A second-order ESDIRK method with adaptive time stepping is applied to stabilize physically insignificant fast waves and accurately integrate the Euler equations in time. The nonlinear system arising at each time step is solved by using a Jacobian-free NewtonKrylov-Schwarz algorithm. To accelerate the convergence and improve the robustness, we employ a class of additive Schwarz preconditioners in which the subdomain Jacobian matrix is constructed using a first-order spatial discretization. Several test cases are used to validate the correctness of the scheme and examine the performance of the solver. Large-scale results on a supercomputer with up to 18, 432 processor cores are provided to show the parallel performance of the proposed method.

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

“…The initial perturbation thus propagates symmetrically to the left and right. Figure 4 shows the cross-sectional potential temperature perturbation at an altitude of y = 5, 000 m. The reference solution is obtained by a spectral-element solver 9 (that solves the governing equations expressed in terms of mass, momentum, and potential temperature as perturbations around the hydrostatically balanced state) using 10th-order polynomials and an effective grid resolution of 250 m. Excellent agreement is observed between the computed solutions and the reference solution, as well as with results in the literature 18,51,21,24 D. Rising Thermal Bubble…”

“…(25), is fifth-order accurate when the solution is smooth (ω k → c k ) and reduces to Eq. (24). Across and near discontinuities, the weights corresponding to the stencils containing the discontinuity approach zero, and a biased (away from the discontinuity) compact scheme is obtained.…”

“…23 High-order finitevolume 8,18,19,21,24 and finite-element or spectral-element 9, 10, 23 methods have been proposed that address these drawbacks. These flows are often characterized by strong gradients, and linear spatial discretization methods often need an additional filter or artificial diffusion to stabilize the solution.…”

Ultra-Scalable CPU-MIC Acceleration of Mesoscale Atmospheric Modeling on Tianhe-2

“…To minimize roundoff errors, values of r 0 ¼ r À r, ðruÞ 0 ¼ ru À ru and p 0 ¼ p À p have been shifted according to the hydrostatic state that satisfies @p @z ¼ Àrg. To solve the Euler equations (1), (2), (3), we make use of a terrain-following mesh and employ a cell-centered finite volume scheme for spatial discretization together with a second-order TVD Runge-Kutta method for time stepping; a similar work has been done in [31] for the Euler equations in 2D. At each time step, two stencil sweeps are applied consecutively at all mesh elements.…”

Addressing the Challenges of Implementation of High-Order Finite-Volume Schemes for Atmospheric Dynamics on Unstructured Meshes

“…For the atmospheric dynamics there is also an overwhelming number of recent state-ofthe-art approaches that utilise high-order schemes for either structured grid or unstructured dynamical cores [4,7,14,15,24,30,32,44,[59][60][61]. Regarding the high-order schemes (higher than 3rd order of spatial accuracy) for unstructured dynamical cores all of the approaches are based on the Discontinuous Galerkin framework and the Lagrange-Galerkin Spectral Element Method [7,15,26,32,60].…”