Multi-dimensional hybrid Fourier continuation–WENO solvers for conservation laws

Shahbazi, Khosro; Hesthaven, Jan S.; Zhu, Xueyu

doi:10.1016/j.jcp.2013.07.009

Cited by 16 publications

(16 citation statements)

References 32 publications

(54 reference statements)

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“…The FC-Gram method, on the other hand, has been used successfully in a wide variety of highly accurate and robust PDEs solvers [3,4,8,11,20,22,23]. The secret of FC-Gram's success appears to be related to the delicate balance between accuracy and stability in solvers for initial boundary-value PDEs.…”

Section: The Fc-gram Approachmentioning

confidence: 99%

“…In practice, Principle 1 has allowed the development of the variety of PDEs solvers presented in [3,4,8,11,20,22,23], which have formally polynomial convergence order that can be observed at domain boundaries, but which also have spectral-like behavior (e.g., exceptionally low dispersion errors) within the interior of the computational domain. (The second example of section 5 demonstrates the spectral nature of the method within the interior.)…”

Section: Accuracy and Convergence Ordermentioning

confidence: 99%

“…, N. This sequence can be interpreted as the samples of a smooth functionf withf j =f (x j ), where x j is defined in (1.4). The FC-Gram extension presented in [3,4,8,11,20,22,23] depends only on some number d of "near-boundary" function values…”

Section: Smooth Extensionmentioning

confidence: 99%

“…Because the intent of the current construction is to reproduce as faithfully as possible the behavior of the FC-Gram approach, we now seek a solution to the problem with the choice M = 25, which is the number of extension points produced by the algorithm employed in [3,4,8,11,20,22,23] …”

Section: An Optimization Problemmentioning

confidence: 99%

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Discrete Periodic Extension Using an Approximate Step Function

Albin¹,

Pathmanathan²

2014

SIAM J. Sci. Comput.

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The discrete periodic extension is a technique for augmenting a given set of uniformly spaced samples of a smooth function with auxiliary values in an extension region. If a suitable extension is constructed, the interpolating trigonometric polynomial found via an FFT will accurately approximate the original function in its original interval of definition. The discrete periodic extension is a key construction in the algorithm FC-Gram (Fourier continuation based on Gram polynomials) algorithm. The FC-Gram algorithm, in turn, lies at the heart of several recent efficient and highorder-accurate PDE solvers. This paper presents a new flexible discrete periodic extension procedure that performs at least as well as the FC-Gram method, but with somewhat simpler constructions and significantly decreased setup time.1. Introduction. The purpose of this paper is to offer a straightforward approach for addressing the approximation problem illustrated in Figure 1. In the figure, the solid curve represents a smooth function f (x) on the interval [0, 1], and the solid circular dots represent N samples of f at uniformly spaced nodes in this interval. The problem addressed in this paper is summarized as follows: Main problem: By using only the first d and last d data points, how does one produce an additional M function values in an interval [1, b] with the property that the trigonometric polynomial interpolant of all N + M samples on [0, b] provides a highly accurate approximation of f (x) in the interval [0, 1]? In the figure, the trigonometric polynomial interpolant is represented by the union of the solid and dashed curves. For illustration purposes, the sizes of N , M , and d in the schematic are smaller than the values for these numbers used in practice.In the numerical comparisons presented in section 5, for example, M = 25, d = 10, and N ≥ 100. The algorithm presented can be viewed conceptually as a method for efficiently producing, for any choice of N ≥ d, a sparse (N +M )×N matrix E per which maps the N samples of the original function to the N + M samples of its periodic extension. The sparseness of the matrix operator derives from the fact that only 2d samples are used to produce the extension, regardless the size of N . As will be shown (see (2.8)), E per has a very simple structure with only N + 2d 2 nonzero entries.Such a matrix, E per , can be quite useful in practice, as it allows the high-precision approximation of a nonperiodic function f and its derivatives by means of FFT-based interpolation of the extended vector. Moreover, due to the sparsity of E per and the efficiency of the FFT, the resulting algorithm will have O(N log N ) complexity. The

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Section: The Fc-gram Approachmentioning

confidence: 99%

Section: Accuracy and Convergence Ordermentioning

confidence: 99%

Section: Smooth Extensionmentioning

confidence: 99%

Section: An Optimization Problemmentioning

confidence: 99%

See 2 more Smart Citations

Discrete Periodic Extension Using an Approximate Step Function

Albin¹,

Pathmanathan²

2014

SIAM J. Sci. Comput.

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“…The natural choice of coupling central difference linear method with WENO scheme (CD-WENO), though stable, suffers from numerical dispersion unsuitable for capturing shock-induced transition and turbulence with physical dispersal of momentum and material as their core characteristics. 21 Another choice of coupling spectral methods with WENO schemes results in non-uniform Chebychev grid points for the spectral domains which does not conform in any natural fashion with the uniform grid of WENO scheme, requiring interpolations for data transfer from uniform to Chebychev points, in turn, potentially yielding unreliable shock detection schemes. 5 Moreover, the spectral-WENO scheme possesses a CFL condition that scales as O(1/N 2 ), much smaller than the pure WENO scheme with O(1/N ).…”

Section: Introductionmentioning

confidence: 99%

High-Order Hybrid Fourier Continuation-WENO Scheme for 3D Compressible Navier-Stokes Equations

Shahbazi

2016

46th AIAA Fluid Dynamics Conference

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This paper presents a parallel multi-domain high-order hybrid scheme, FC-WENO, arising from the conjugation of Fourier-continuation (FC) method and weighted essentially nonoscillatory scheme (WENO) for three-dimensional unsteady compressible Navier-Stokes equations. The hybrid FC-WENO scheme with almost dispersion-less and dissipation-less resolutions (away from the shock waves) outperforms alternatives based on pure WENO scheme and central difference/WENO (CD-WENO) hybrid combination. The efficacy of FC-WENO scheme is demonstrated through computations of broadband motion of highspeed isotropic turbulence with eddy shocklets, whereby on the identical grid and with comparable CPU times, the FC-WENO results in more accurate average turbulence measures as compared to those of pure WENO and CD-WENO schemes.

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A New Framework of GPU-Accelerated Spectral Solvers: Collocation and Glerkin Methods for Systems of Coupled Elliptic Equations

Chen

2014

J Sci Comput

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Multi-dimensional hybrid Fourier continuation–WENO solvers for conservation laws

Cited by 16 publications

References 32 publications

Discrete Periodic Extension Using an Approximate Step Function

Discrete Periodic Extension Using an Approximate Step Function

High-Order Hybrid Fourier Continuation-WENO Scheme for 3D Compressible Navier-Stokes Equations

A New Framework of GPU-Accelerated Spectral Solvers: Collocation and Glerkin Methods for Systems of Coupled Elliptic Equations

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