A new category of Hermitian upwind schemes for computational acoustics

Capdeville, G.

doi:10.1016/j.jcp.2005.04.005

Cited by 6 publications

(50 citation statements)

References 22 publications

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“…In [1], we demonstrated that the -P3 scheme only needs 15 CPW to reach an accuracy of 0.01% of error in phase and in amplitude. With such a result, the -P3 scheme appears to be more advantageous (both in CPU time and accuracy) than classical finite-difference methods, even simpler formally.…”

Section: Introductionmentioning

confidence: 91%

“…In Section 3, we extend the -P5 and the RGK-P5 schemes to linear hyperbolic systems, following the procedures developed in [1,3]. We propose two solutions for the evolution stage in order to obtain the algebraical expressions that characterize the -P5 and the RGK-P5 scheme.…”

Section: G Capdevillementioning

confidence: 99%

“…We propose two solutions for the evolution stage in order to obtain the algebraical expressions that characterize the -P5 and the RGK-P5 scheme. The boundary conditions are briefly detailed, according to the principles explained in [1,3]. Section 4 presents extensive numerical simulation in order to validate the boundary conditions and to compare both schemes.…”

Section: G Capdevillementioning

confidence: 99%

“…To begin, we describe the construction of the -P5 scheme in order to discretize (1). We define such a scheme by the choice of the biased stencil [i −1, i] and by the following discrete unknowns:…”

Section: Fifth-order Upwind Hermitian Schemes: One-dimensional Casementioning

confidence: 99%

“…Such a method must preserve the compactness of the discretization while necessitating less than 8 CPW to reach an accuracy of 0.01% of error in phase and in amplitude (roughly the performances of the DRP scheme [5], see [1] for a detailed analysis). For this purpose, we rely on the algorithm developed to generate the -P3 scheme [1,3].…”

Section: Introductionmentioning

confidence: 99%

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Fifth‐order Hermitian schemes for computational linear aeroacoustics

Capdeville

2007

Numerical Methods in Fluids

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SUMMARYWe develop a class of fifth-order methods to solve linear acoustics and/or aeroacoustics. Based on local Hermite polynomials, we investigate three competing strategies for solving hyperbolic linear problems with a fifth-order accuracy. A one-dimensional (1D) analysis in the Fourier series makes it possible to classify these possibilities. Then, numerical computations based on the 1D scalar advection equation support two possibilities in order to update the discrete variable and its first and second derivatives: the first one uses a procedure similar to that of Cauchy-Kovaleskaya (the ' -P5 scheme'); the second one relies on a semi-discrete form and evolves in time the discrete unknowns by using a five-stage Runge-Kutta method (the 'RGK-P5 scheme'). Although the RGK-P5 scheme shares the same local spatial interpolator with the -P5 scheme, it is algebraically simpler. However, it is shown numerically that its loss of compactness reduces its domain of stability.Both schemes are then extended to bi-dimensional acoustics and aeroacoustics. In the 'reconstruction stage', we define a fifth-order local spatial interpolator based on an upwind stencil. In the 'decomposition stage', we decompose the time derivatives into simple wave contributions.In the 'evolution stage', we use these fluctuations to update either by a Cauchy-Kovaleskaya procedure or by a five-stage Runge-Kutta algorithm, the discrete variable and its derivatives.In this way, depending on the configuration of the 'evolution stage', two fifth-order upwind Hermitian schemes are constructed.The effectiveness and the exactitude of both schemes are checked by their applications to several 2D problems in acoustics and aeroacoustics. In this aim, we compare the computational cost and the computation memory requirement for each solution.The RGK-P5 appears as the best compromise between simplicity and accuracy, while the -P5 scheme is more accurate and less CPU time consuming, despite a greater algebraic complexity.

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Section: Introductionmentioning

confidence: 91%

Section: G Capdevillementioning

confidence: 99%

Section: G Capdevillementioning

confidence: 99%

Section: Fifth-order Upwind Hermitian Schemes: One-dimensional Casementioning

confidence: 99%