2018
DOI: 10.1007/s10915-018-0719-5
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Shock Regularization with Smoothness-Increasing Accuracy-Conserving Dirac-Delta Polynomial Kernels

Abstract: A smoothness-increasing accuracy conserving filtering approach to the regularization of discontinuities is presented for single domain spectral collocation approximations of hyperbolic conservation laws. The filter is based on convolution of a polynomial kernel that approximates a delta-sequence. The kernel combines a k th order smoothness with an arbitrary number of m zero moments. The zero moments ensure a m th order accurate approximation of the delta-sequence to the delta function. Through exact quadrature… Show more

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Cited by 10 publications
(17 citation statements)
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“…To obtain accurate solutions and consistency between MC and the CDF equation, we must be careful in selecting the numerical methods that we use to approximate the governing systems. Here, we rely on a low dispersive and low diffusive, single domain Chebyshev collocation method and use some of the recently developed filtering and regularization techniques to capture shocks and regularize sources [26,27].…”
Section: Methodsmentioning
confidence: 99%
“…To obtain accurate solutions and consistency between MC and the CDF equation, we must be careful in selecting the numerical methods that we use to approximate the governing systems. Here, we rely on a low dispersive and low diffusive, single domain Chebyshev collocation method and use some of the recently developed filtering and regularization techniques to capture shocks and regularize sources [26,27].…”
Section: Methodsmentioning
confidence: 99%
“…In Fig. 1 we illustrate the polynomial approximation of the delta kernel According to the SIAC filtering strategy [29,34,37] we regularize the solution produced by the numerical scheme with the manipulation (3.5) For compact notation we introduce the filter matrix Φ and approximate its values with LGL quadrature by mapping the corresponding integration area…”
Section: Siac Filtermentioning
confidence: 99%
“…Thus, oscillations caused by shocks as well as by re-interpolation (Runge phenomena) cannot be smoothed in these areas. In the original approach for the global collocation the affected parts of the discretization are set to the analytical solution [37]. Using a local version of the filter we can avoid identifying interior points by an analytical reference solution, as discussed in the next section.…”
Section: Siac Filtermentioning
confidence: 99%
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“…We subsequently solve the deterministic and stochastic linear advection equation using the Chebyshev collocation method [27] in space. For the time discretization we used a fourth-order Runge-Kutta method.…”
Section: Numerical Approximationsmentioning
confidence: 99%