A scale-selective multilevel method for long-wave linear acoustics

Vater, Stefan; Klein, Rupert; Knio, Omar M.

doi:10.2478/s11600-011-0037-x

Cited by 9 publications

(12 citation statements)

References 15 publications

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“…In contrast, by introducing multigrid decompositions of the flow and a smooth blending of time integrators, we obtain a scheme in this work that allows for much more general data with true multiscale content. Our work extends that of [42] from linear wave propagation in one space dimension to the nonlinear shallow water equations.…”

Section: Introductionsupporting

confidence: 53%

“…This construction ensures that the discretization correctly approximates the limit behavior of the equations. A second ingredient is a scale-selective multilevel scheme which was previously derived for the linearized equations [42,40]. With this addition we account for the characteristic flow behavior on the different scales resolved by the discretization.…”

Section: Numerical Schemementioning

confidence: 99%

“…Multiscale scheme. With the introduction of the trapezoidal and the BDF(2)based time discretizations for the second correction, all ingredients are now at hand to apply the multilevel scheme from Vater et al [42] as part of a semi-implicit method to the fully nonlinear shallow water equations. The idea is to define direct scale dependent splittings of the fields for fluid depth and momentum, i.e.,…”

Section: Second Correctionmentioning

confidence: 99%

“…To achieve these goals, a semi-implicit method for the nonlinear shallow water equations is combined with a multilevel approach which has successfully been applied to the linearized equations to model multiscale behavior in Vater et al [42]. The latter enables the association of different solution components with certain spatial scales and is based on geometric multigrid ideas.…”

Section: Introductionmentioning

confidence: 99%

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A semi-implicit multiscale scheme for shallow water flows at low Froude number

Vater

Klein

2018

Commun. Appl. Math. Comput. Sci.

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A new large time step semi-implicit multiscale method is presented for the solution of low Froude-number shallow water flows. While on small scales which are under-resolved in time the impact of source terms on the divergence of the flow is essentially balanced, on large resolved scales the scheme propagates free gravity waves with minimized diffusion. The scheme features a scale decomposition based on multigrid ideas. Two different time integrators are blended at each scale depending on the scale-dependent Courant number for gravity wave propagation. The finite-volume discretization is based on a Cartesian grid and is second order accurate. The basic properties of the method are validated by numerical tests. This development is a further step in the development of asymptotically adaptive numerical methods for the computation of large scale atmospheric flows.

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Section: Introductionsupporting

confidence: 53%

Section: Numerical Schemementioning

confidence: 99%

Section: Second Correctionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A semi-implicit multiscale scheme for shallow water flows at low Froude number

Vater

Klein

2018

Commun. Appl. Math. Comput. Sci.

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“…Acoustic-advection with multi-scale initial data. To assess how well fwsw-SDC damps highly oscillatory modes, we study an example from Vater et al [40] with multi-scale initial data. Let Figure 6 shows the solution produced by SDC, DIRK and IMEX methods of order two (left) and four (right).…”

Section: 2mentioning

confidence: 99%

Spectral Deferred Corrections with Fast-wave Slow-wave Splitting

Ruprecht¹,

Speck²

2016

SIAM J. Sci. Comput.

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Abstract. The paper investigates a variant of semi-implicit spectral deferred corrections (SISDC) in which the stiff, fast dynamics correspond to fast propagating waves ("fast-wave slow-wave problem"). We show that for a scalar test problem with two imaginary eigenvalues iλ f , iλs, having ∆t (|λ f | + |λs|) < 1 is sufficient for the fast-wave slow-wave SDC (fwsw-SDC) iteration to converge and that in the limit of infinitely fast waves the convergence rate of the non-split version is retained. Stability function and discrete dispersion relation are derived and show that the method is stable for essentially arbitrary fast-wave CFL numbers as long as the slow dynamics are resolved. The method causes little numerical diffusion and its semi-discrete phase speed is accurate also for large wave number modes. Performance is studied for an acoustic-advection problem and for the linearised Boussinesq equations, describing compressible, stratified flow. fwsw-SDC is compared to a diagonally implicit Runge-Kutta (DIRK) and IMEX Runge-Kutta (IMEX) method and found to be competitive in terms of both accuracy and cost.Key words. spectral deferred corrections, fast-wave slow-wave splitting, Euler equations, acoustic-advection AMS subject classifications.1. Introduction. For simulations of compressible flow, in particular in numerical weather prediction and climate simulations, the presence of acoustic waves can pose significant numerical challenges to the time integration method. Explicit methods are restricted to inefficiently small steps while fully implicit methods are expensive and can artificially slow down high wave number modes. The fully compressible equations can be replaced by filtered models that do not support sound waves [9,30], but these require solution of a Poisson problem in each step and have difficulties capturing large-scale wave dynamics [8].Therefore, a widely used class of methods are split-explicit integrators: they separate the equation into fast and slow processes which are then integrated with different time step sizes and different (explicit) methods. A popular method of this type is a third-order Runge-Kutta scheme combined with a forward-backward Euler integrator for the acoustic terms [45]. While computationally efficient, split-explicit methods typically require some form of damping for stabilisation [3], which reduces their effective order of accuracy. However, a second-order split-explicit two-step peer method has recently been derived that allows for stable integration of the compressible Euler equations without damping [23].Another form of splitting are semi-implicit methods. They also split the equations into fast and slow parts but then use an implicit method for the fast and an explicit method for the slow part. In many applications, the fast, stiff terms stem from diffusion and/or rapid chemical reactions and methods with IMEX splitting for equations of reaction-diffusion type have been widely studied [24,29,34]. For stratified, compressible flows, however, the fast dynamics are not dif...

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Multiscale Numerical Methods in Atmospheric Science

Klein¹

2015

Encyclopedia of Applied and Computational Mathematics

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A scale-selective multilevel method for long-wave linear acoustics

Cited by 9 publications

References 15 publications

A semi-implicit multiscale scheme for shallow water flows at low Froude number

A semi-implicit multiscale scheme for shallow water flows at low Froude number

Spectral Deferred Corrections with Fast-wave Slow-wave Splitting

Multiscale Numerical Methods in Atmospheric Science

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