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11
Implicit–Explicit Multistep Methods for FastWave–SlowWave Problems
, 2011
"... Implicit–explicit (IMEX) linear multistep methods are examined with respect to their suitability for the integration of fastwave–slowwave problems in which the fast wave has relatively low amplitude and need not be accurately simulated. The widely used combination of trapezoidal implicit and leapf ..."
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Implicit–explicit (IMEX) linear multistep methods are examined with respect to their suitability for the integration of fastwave–slowwave problems in which the fast wave has relatively low amplitude and need not be accurately simulated. The widely used combination of trapezoidal implicit and leapfrog explicit differencing is compared to schemes based on Adams methods or on backward differencing. Two new families of methods are proposed that have good stability properties in fastwave–slowwave problems: one family is based on Adams methods and the other on backward schemes. Here the focus is primarily on four specific schemes drawn from these two families: a pair of Adams methods and a pair of backward methods that are either (i) optimized for thirdorder accuracy in the explicit component of the full IMEX scheme, or (ii) employ particularly good schemes for the implicit component. These new schemes are superior, in many respects, to the linear multistep IMEX schemes currently in use. The behavior of these schemes is compared theoretically in the context of the simple oscillation equation and also for the linearized equations governing stratified compressible flow. Several schemes are also tested in fully nonlinear simulations of gravity waves generated by a localized source in a shear flow. 1.
Adaptive discontinuous evolution Galerkin method for dry atmospheric flow
 J. Comput. Phys
, 2014
"... We present a new adaptive genuinely multidimensional method within the framework of the discontinuous Galerkin method. The discontinuous evolution Galerkin (DEG) method couples a discontinuous Galerkin formulation with approximate evolution operators. The latter are constructed using the bicharact ..."
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We present a new adaptive genuinely multidimensional method within the framework of the discontinuous Galerkin method. The discontinuous evolution Galerkin (DEG) method couples a discontinuous Galerkin formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of multidimensional hyperbolic systems, such that all of the infinitely many directions of wave propagation are considered explicitly. In order to take into account multiscale phenomena that typically appear in atmospheric flows nonlinear fluxes are split into a linear part governing the acoustic and gravitational waves and to the rest nonlinear part that models advection. Time integration is realized by the IMEX type approximation using the semiimplicit secondorder backward differentiation formulas (BDF2) scheme. Moreover in order to approximate efficiently small scale phenomena adaptive mesh refinement using the space filling curves via AMATOS function library is applied. Three standard meteorological test cases are used to validate the new discontinuous evolution Galerkin method for dry atmospheric convection. Comparisons with the standard onedimensional approximate Riemann solver used for the flux integration demonstrate better stability, accuracy as well as reliability of the new multidimensional DEG method. Key words: dry atmospheric convection, steady states, systems of hyperbolic balance laws, Euler equations, large time step, semiimplicit approximation, evolution Galerkin schemes
Unstructured HighOrder GalerkinTemporalBoundary Methods for the KleinGordon Equation with NonReflecting Boundary Conditions
, 2010
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AN ADAPTIVE DISCONTINUOUS GALERKIN METHOD FOR MODELING ATMOSPHERIC CONVECTION
, 2011
"... Abstract. Theoretical understanding and numerical modeling of atmospheric moist convection still pose great challenges to meteorological research. A Direct Numerical Simulation of a single cumulus cloud is beyond the capacity of today’s computing power. The use of a Large Eddy Simulation in combina ..."
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Abstract. Theoretical understanding and numerical modeling of atmospheric moist convection still pose great challenges to meteorological research. A Direct Numerical Simulation of a single cumulus cloud is beyond the capacity of today’s computing power. The use of a Large Eddy Simulation in combination with semiimplicit timeintegration and adaptive techniques offers a significant reduction of complexity. This paper presents a first step towards an efficient simulation of a single cloud. So far this work is restricted to dry flow in twodimensional geometry without subgridscale modeling. The compressible Euler equations are discretized using a discontinuous Galerkin method introduced by Giraldo and Warburton in 2008. Time integration is done by a semiimplicit backward difference formula. This paper represents the first application of a triangular discontinuous Galerkin method with hadaptive grid refinement for nonhydrostatic atmospheric flow. This refinement of our triangular grid is implemented with the function library AMATOS and uses an efficient space filling curve approach. Validation through different test cases shows very good agreement between the current results and those from the literature. For comparing different adaptivity setups we developed a new qualitative error measure for the simulation of warm air bubbles. With the help of this criterion we show that the simulation of a rising warm air bubble on a locally refined grid can be four times faster than a similar computation on a uniform mesh while still producing the same accuracy. Remarkably only 5 % of the total CPU time is used for adapting the grid after each timestep. 1
Development of the Nonhydrostatic Unified Model of the Atmosphere (NUMA): LimitedArea Mode
"... This paper describes a Nonhydrostatic Unified Model of the Atmosphere (NUMA) based on a spectral element, or highorder continuous Galerkin (CG) spatial discretization utilizing 3D hexahedral elements. The nonhydrostatic dynamical core, based on the compressible Euler equations, is appropriate for ..."
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This paper describes a Nonhydrostatic Unified Model of the Atmosphere (NUMA) based on a spectral element, or highorder continuous Galerkin (CG) spatial discretization utilizing 3D hexahedral elements. The nonhydrostatic dynamical core, based on the compressible Euler equations, is appropriate for both limitedarea and global atmospheric simulations. In this paper, we restrict our attention to 3D limitedarea phenomena; global atmospheric simulations will be presented in a followup paper. A suite of explicit and semiimplicit timeintegrators is presented. Domain decomposition and communication algorithms utilized by our distributed memory implementation is presented, allowing efficient evaluation of the the direct stiffness summation (DSS) operator. Numerical verification of the model is performed using four test cases: 1) 2D inertiagravity waves, 2) flow past a 3D linear hydrostatic mountain, 3) flow past a 3D nonlinear mountain and 4) 3D buoyant convection of a bubble in a neutral atmosphere; these tests indicate that NUMA can simulate the necessary physics of a dry numerical weather prediction dynamical core. Scalability for the explicit dynamical core is demonstrated for 12288 cores on TACC’s Ranger cluster, while the semiimplicit core is shown to scale to 4096 cores on the same architecture.
Comparison between adaptive and uniform discontinuous Galerkin simulations in dry 2D bubble experiments
"... Adaptive mesh refinement generally aims to increase computational efficiency without compromising the accuracy of the numerical solution. However it is an open question in which regions the spatial resolution can actually be coarsened without affecting the accuracy of the result. This question is in ..."
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Adaptive mesh refinement generally aims to increase computational efficiency without compromising the accuracy of the numerical solution. However it is an open question in which regions the spatial resolution can actually be coarsened without affecting the accuracy of the result. This question is investigated for a specific example of dry atmospheric convection, namely the simulation of warm air bubbles. For this purpose a novel numerical model is developed that is tailored towards this specific application. The compressible Euler equations are solved with a Discontinuous Galerkin method. Time integration is done with an IMEXmethod and the dynamic grid adaptivity uses space filling curves via the AMATOS function library. So far the model is able to simulate dry flow in twodimensional geometry without subgridscale modeling. The model is tested with three standard test cases. An error indicator is introduced for a warm air bubble test case which allows one to compare the accuracy between different choices of refinement regions without knowing the exact solution. Essentially this is done by comparing features of the solution that are strongly sensitive to spatial resolution. For the rising warm air bubble the additional error by using adaptivity is smaller than 1 % of the total numerical error if the average number of elements used for the adaptive simulation is about a factor of two times smaller than the number used for the simulation with the uniform fineresolution grid. Correspondingly the adaptive simulation is almost two times faster than the uniform simulation. Furthermore the adaptive simulation is more accurate than a uniform simulation when both use the same CPUtime.
simulations in dry 2D bubble experiments
, 2012
"... Comparison between adaptive and uniform ..."
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A POLYNOMIALBASED NONLINEAR LEAST SQUARES OPTIMIZED PRECONDITIONER FOR CONTINUOUS AND DISCONTINUOUS ELEMENTBASED DISCRETIZATIONS OF THE EULER EQUATIONS
"... Abstract. We introduce a method for constructing a polynomial preconditioner using a nonlinear least squares (NLLS) algorithm. We show that this polynomialbased NLLSoptimized (PBNO) preconditioner significantly improves the performance of 2D continuous Galerkin (CG) and discontinuous Galerkin (D ..."
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Abstract. We introduce a method for constructing a polynomial preconditioner using a nonlinear least squares (NLLS) algorithm. We show that this polynomialbased NLLSoptimized (PBNO) preconditioner significantly improves the performance of 2D continuous Galerkin (CG) and discontinuous Galerkin (DG) fluid dynamical research models when run in an implicitexplicit time integration mode. When employed in a serially computed Schurcomplement form of the 2D CG model with positive definite spectrum, the PBNO preconditioner achieves greater reductions in GMRES iterations and model wallclock time compared to the analogous linear leastsquaresderived Chebyshev polynomial preconditioner. Whereas constructing a Chebyshev preconditioner to handle the complex spectrum of the DG model would introduce an element of arbitrariness in selecting the appropriate convex hull, construction of a PBNO preconditioner for the 2D DG model utilizes precisely the same objective NLLS algorithm as for the CG model. As in the CG model, the PBNO preconditioner achieves significant reduction in GMRES iteration counts and model wallclock time. Comparisons of the ability of the PBNO preconditioner to improve CG and DG model performance when employing the Stabilized Biconjugate Gradient algorithm (BICGS) and the basic Richardson (RICH) iteration are also included. In particular, we show that higher order PBNO preconditioning of the Richardson iteration (which is run in a dot product free mode) makes the algorithm competitive with GMRES and BICGS in a serial computing environment, especially when employed in a DG model. Because the NLLSbased
AN ELEMENTBASED SPECTRALLYOPTIMIZED APPROXIMATE INVERSE PRECONDITIONER FOR THE EULER EQUATIONS
, 2012
"... approximate inverse preconditioner for ..."
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