Parallel multilevel methods for implicit solution of shallow water equations with nonsmooth topography on the cubed-sphere

Yang, Chao; Cai, Xiao‐Chuan

doi:10.1016/j.jcp.2010.12.027

Cited by 30 publications

(23 citation statements)

References 48 publications

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“…As a result, this test has been controversial in its usefulness [7,26]. Even so, it is still commonly used as a test case for novel numerical schemes in atmospheric modeling [7,30,35,48]. However, it should be noted that these instabilities necessitates the use of a filter, regardless of the numerical scheme.…”

Section: Rossby-haurwitz Wavesmentioning

confidence: 99%

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

Flyer

Lehto

Blaise

et al. 2012

Journal of Computational Physics

160

173

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. AbstractThe current paper establishes the computational efficiency and accuracy of the RBF-FD method for large-scale geoscience modeling with comparisons to state-of-the-art methods as high-order discontinuous Galerkin and spherical harmonics, the latter using expansions with close to 300,000 bases. The test cases are demanding fluid flow problems on the sphere that exhibit numerical challenges, such as Gibbs phenomena, sharp gradients, and complex vortical dynamics with rapid energy transfer from large to small scales over short time periods. The computations were possible as well as very competitive due to the implementation of hyperviscosity on large RBF stencil sizes (corresponding roughly to 6th to 9th order methods) with up to O(10 5 ) nodes on the sphere. The RBF-FD method scaled as O(N ) per time step, where N is the total number of nodes on the sphere. In the Appendix, guidelines are given on how to chose parameters when using RBF-FD to solve hyperbolic PDEs.

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Section: Rossby-haurwitz Wavesmentioning

confidence: 99%

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

Flyer

Lehto

Blaise

et al. 2012

Journal of Computational Physics

160

173

View full text Add to dashboard Cite

show abstract

“…Although several studies have been carried out for these schemes for some linear problems, it is not clear which one is optimal for the Euler equations. In [48], κ = 0 is used for the global shallow water equations, while in [43], κ = 1/3 is used for the vertical discretization of the compressible Euler equations. In our finite volume scheme, we choose κ = 1/2 because of its low numerical dissipation as observed in our numerical experiments.…”

Section: Ausmmentioning

confidence: 99%

“…6.3]) and δX n is the Newton correction obtained by solving the Jacoian system as discussed later. To achieve a more uniform distribution of residual errors of all time steps [48], the stopping condition for the Newton iteration (5.1) is adaptively determined by…”

Section: Fully Implicit Adaptive Time Steppingmentioning

confidence: 99%

“…However, at each time step, due to the need of solving a nonlinear system, a fully implicit method might be much more expensive than an explicit or an operator splitting method. Despite of some successes on computing fully implicit solutions of the global shallow water equations [10,11,48,49], it is rare to see the application of fully implicit methods in nonhydrostatic atmospheric simulations. One of the contributions is [40], in which a discontinuous Galerkin solver for mesoscale flows using a fully implicit Rosenbrock scheme is proposed and shown to be superior to an explicit Runge-Kutta method.…”

mentioning

confidence: 99%

“…On the other hand, we have made some efforts on employing domain decomposition based algorithms in the fully implicit solution of the global shallow water equations [48]. The solver has been shown in several numerical experiments to be highly scalable to tens of thousands of processors in terms of both strong and weak scalabilities.…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Yang¹,

Cai²

2014

SIAM J. Sci. Comput.

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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.

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A parallel non‐nested two‐level domain decomposition method for simulating blood flows in cerebral artery of stroke patient

Chen

Cheng

et al. 2020

Numer Methods Biomed Eng

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Numerical simulation of blood flows in patient‐specific arteries can be useful for the understanding of vascular diseases, as well as for surgery planning. In this paper, we simulate blood flows in the full cerebral artery of stroke patients. To accurately resolve the flow in this rather complex geometry with stenosis is challenging and it is also important to obtain the results in a short amount of computing time so that the simulation can be used in pre‐ and/or post‐surgery planning. For this purpose, we introduce a highly scalable, parallel non‐nested two‐level domain decomposition method for the three‐dimensional unsteady incompressible Navier‐Stokes equations with an impedance outlet boundary condition. The problem is discretized with a stabilized finite element method on unstructured meshes in space and a fully implicit method in time, and the large nonlinear systems are solved by a preconditioned parallel Newton‐Krylov method with a two‐level Schwarz method. The key component of the method is a non‐nested coarse problem solved using a subset of processor cores and its solution is interpolated to the fine space using radial basis functions. To validate and verify the proposed algorithm and its highly parallel implementation, we consider a case with available clinical data and show that the computed result matches with the measured data. Further numerical experiments indicate that the proposed method works well for realistic geometry and parameters of a full size cerebral artery of an adult stroke patient on a supercomputers with thousands of processor cores.

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Parallel multilevel methods for implicit solution of shallow water equations with nonsmooth topography on the cubed-sphere

Cited by 30 publications

References 48 publications

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

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

A parallel non‐nested two‐level domain decomposition method for simulating blood flows in cerebral artery of stroke patient

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