A spectral element solution of the Klein–Gordon equation with high-order treatment of time and non-reflecting boundary

Lindquist, Joseph M.; Neta, Beny; Giraldo, Francis X.

doi:10.1016/j.wavemoti.2009.11.007

Cited by 16 publications

(13 citation statements)

References 27 publications

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“…Subsequent spectral element and boundary work by the current authors [20] using the G-N formulation of the dispersive wave equation on a semi-infinite channel showed similar results to those presented by Kucherov and Givoli and how a highorder treatment of the time domain (up to order 10) produces additional improvements. These improvements, however, have their limits thus confirming the hypothesis of Kucherov and Givoli.…”

Section: Introductionsupporting

confidence: 74%

“…Even for high order (order 8 and 16) spectral elements, the gains made by increasing the order of the NRBC are limited at some point using RK4. In [20] the authors showed that high-order time integration allowed boundary gains to improve solution quality for the KGE under zero advection. For this experiment, consider the KGE on a semi-infinite domain with h = 0 on C W .…”

Section: Effects Of Time Integration Techniquementioning

confidence: 99%

“…It should be noted that the only other spectral element, high-order boundary approach are Kucherov and Givoli [19] and Lindquist et al [20,13]. Kucherov and Givoli demonstrate exponential error convergence of the classical wave equation on a semi-infinite channel when solved using spectral elements and high order boundary treatment (using the H-W boundary scheme).…”

Section: Introductionmentioning

confidence: 99%

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High-order non-reflecting boundary conditions for dispersive waves in polar coordinates using spectral elements

Lindquist¹,

Neta²,

Giraldo³

2012

Applied Mathematics and Computation

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Section: Introductionsupporting

confidence: 74%

Section: Effects Of Time Integration Techniquementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High-order non-reflecting boundary conditions for dispersive waves in polar coordinates using spectral elements

Lindquist¹,

Neta²,

Giraldo³

2012

Applied Mathematics and Computation

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“…In particular, the sponge layer makes it impossible to build a model that is conservative with respect to fluxes prescribed on the boundary. To overcome this problem, we have begun work on the construction of high-order NRBCs that can be used with high-order spatial and temporal discretizations (see [6] and [26]) but we are still far away from implementing such methods into Navier-Stokes models. Unfortunately, sponge-based NRBCs are those typically used today in industrial-type nonhydrostatic atmospheric models.…”

Section: Courant Number = Maxmentioning

confidence: 99%

“…This test case emphasizes the need for better NRBCs that are high order accurate and conservative; unfortunately, NRBCs such as the ones we use here are used today in all industrial-type nonhydrostatic atmospheric models. To overcome the first problem (accuracy), we have begun work on the construction of high-order NRBCs that can be used with high-order spatial and temporal discretizations (see [6] and [26]) but we are still far away from implementing such methods into Navier-Stokes models. The second problem is more complicated to overcome.…”

Section: Gmres Iterationsmentioning

confidence: 99%

Semi-Implicit Formulations of the Navier–Stokes Equations: Application to Nonhydrostatic Atmospheric Modeling

Giraldo¹,

Restelli²,

Läuter³

2010

SIAM J. Sci. Comput.

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Abstract. We present semi-implicit (IMEX) formulations of the compressible Navier-Stokes equations (NSE) for applications in nonhydrostatic atmospheric modeling. The compressible NSE in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the Schur complement form of the linear implicit problem. We present results for five different forms of the compressible NSE and describe in detail how to formulate the semi-implicit time-integration method for these equations. Finally, we compare all five equations and compare the semi-implicit formulations of these equations both using the Schur and No Schur forms against an explicit Runge-Kutta method. Our simulations show that, if efficiency is the main criterion, it matters which form of the governing equations you choose. Furthermore, the semi-implicit formulations are faster than the explicit RungeKutta method for all the tests studied especially if the Schur form is used. While we have used the spectral element method for discretizing the spatial operators, the semi-implicit formulations that we derive are directly applicable to all other numerical methods. We show results for our five semi-implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling, including: inertia gravity waves, rising thermal bubbles (i.e., Rayleigh-Taylor instabilities), density current (i.e., Kelvin-Helmholtz instabilities), and mountain test cases; the latter test case requires the implementation of non-reflecting boundary conditions. Therefore, we show results for all five semi-implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling: no-flux (reflecting) and non-reflecting boundary conditions. It is shown that the non-reflecting boundary conditions exert a strong impact on the accuracy and efficiency of the models.Key words. compressible flow; element-based Galerkin methods; Euler; IMEX; Lagrange; Legendre; NavierStokes; nonhydrostatic; spectral elements; time-integration.AMS subject classifications. 65M60, 65M70, 35L65, 86A101. Introduction. It can be argued that the single most important property of an operational nonhydrostatic mesoscale atmospheric is efficiency. Clearly, this efficiency should not come at the cost of accuracy but if a weather center has the choice between a very accurate model and one that is efficient, they will probably pick the efficient one; however, as numerical analysts, we would like to build models that are both accurate and efficient. One way to achieve this goal is to construct numerical models based on high-order methods: this class of methods offers exponential (spectral) convergence for smooth problems and achieves excellent scalability on modern multi-core systems if they are used in an element-based approach (i.e., if the approximating polynomials have compact/local support). This is the idea behind element-based Galerkin methods such as spectral element (SE) and discontinuous Galerkin (DG) methods (see [14] and [2...

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Comparison of high‐order absorbing boundary conditions and perfectly matched layers in the frequency domain

Rabinovich

Givoli

Bécache

2010

Numer Methods Biomed Eng

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SUMMARYThe need for numerical schemes for wave problems in large and unbounded domains appears in various applications, including modeling of pressure waves in arteries and other problems in biomedical engineering. Two powerful methods to handle such problems via domain truncation are the use of high-order absorbing boundary conditions (ABCs) and perfectly matched layers (PMLs). A numerical study is presented to compare the performance of these two types of methods, for two-dimensional problems governed by the Helmholtz equation. The high-order ABCs employed here are of the Hagstrom-Warburton type; they are adapted and applied to the frequency domain for the first time. Four PMLs are examined, with linear, quadratic, constant and unbounded decay functions. Two planar configurations are considered: a waveguide and a quarter plane. In the latter case, special corner conditions are developed and used in conjunction with the ABC. One of the main conclusions from the ABC-PML comparison is that in the high-accuracy regime, the ABC scheme and the unbounded PML are equally effective.

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A spectral element solution of the Klein–Gordon equation with high-order treatment of time and non-reflecting boundary

Cited by 16 publications

References 27 publications

High-order non-reflecting boundary conditions for dispersive waves in polar coordinates using spectral elements

High-order non-reflecting boundary conditions for dispersive waves in polar coordinates using spectral elements

Semi-Implicit Formulations of the Navier–Stokes Equations: Application to Nonhydrostatic Atmospheric Modeling

Comparison of high‐order absorbing boundary conditions and perfectly matched layers in the frequency domain

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