Shock Regularization with Smoothness-Increasing Accuracy-Conserving Dirac-Delta Polynomial Kernels

Wissink, B. W.; Jacobs, Gustaaf; Ryan, Jennifer K.; Don, Wai Sun; Weide, Edwin Theodorus Antonius van der

doi:10.1007/s10915-018-0719-5

Cited by 10 publications

(17 citation statements)

References 37 publications

(60 reference 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%

Method of Distributions for Systems with Stochastic Forcing

Rutjens¹,

Jacobs²,

Tartakovsky³

2019

Preprint

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The method of distributions is developed for systems that are governed by hyperbolic conservation laws with stochastic forcing. The method yields a deterministic equation for the cumulative density distribution (CDF) of a system state, e.g., for flow velocity governed by an inviscid Burgers' equation with random source coefficients. This is achieved without recourse to any closure approximation. The CDF model is verified against MC simulations using spectral numerical approximations. It is shown that the CDF model accurately predicts the mean and standard deviation for Gaussian, normal and beta distributions of the random coefficients.

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Section: Methodsmentioning

confidence: 99%

Method of Distributions for Systems with Stochastic Forcing

Rutjens¹,

Jacobs²,

Tartakovsky³

2019

Preprint

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“…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%

“…Typically, such SIAC filters were designed to obtain super-convergence in a post-processing step by a convolution of the numerical approximation against a specifically designed kernel function once at the final time [5,9,19,26,29,33]. However, recent work has applied the SIAC filter as a shock capturing and/or regularization of general discontinuities strategy during the computation of the approximate solution for global spectral collocation methods [34,37]. For such spectral methods, the SIAC filter is suitable to apply in shocked regions because said filter can recover full accuracy away from shocks, see e.g [19].…”

Section: Introductionmentioning

confidence: 99%

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Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods

et al. 2019

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We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. [37]. In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well known that high-order methods generate spurious oscillations near discontinuities which can develop in the solution for nonlinear problems, even when the initial data is smooth. We propose a novel multi-element SIAC filtering technique applied to the DGSEM as a shock capturing method. We design the SIAC filtering such that the numerical scheme remains high-order accurate and that the shock capturing is applied adaptively throughout the domain. The shock capturing method is derived for general systems of conservation laws. We apply the novel SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics (MHD) equations to several standard test problems with a variety of boundary conditions.

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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%

Uncertainty quantification in Eulerian-Lagrangian simulations of (point-)particle-laden flows with data-driven and empirical forcing models

Fountoulakis¹,

Udaykumar²,

Jacobs³

2018

Preprint

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An uncertainty quantification framework is developed for Eulerian-Lagrangian models of particle-laden flows, where the fluid is modeled through a system of partial differential equations in the Eulerian frame and inertial particles are traced as points in the Lagrangian frame. The source of uncertainty in such problems is the particle forcing, which is determined empirically or computationally with high-fidelity methods (data-driven). The framework relies on the averaging of the deterministic governing equations with the stochastic forcing and allows for an estimation of the first and second moment of the quantities of interest. Via comparison with Monte Carlo simulations, it is demonstrated that the moment equations accurately predict the uncertainty for problems whose Eulerian dynamics are either governed by the linear advection equation or the compressible Euler equations. In areas of singular particle interfaces and shock singularities significant uncertainty is generated. An investigation into the effect of the numerical methods shows that low-dissipative higher-order methods are necessary to capture numerical singularities (shock discontinuities, singular source terms, particle clustering) with low diffusion in the propagation of uncertainty.

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Shock Regularization with Smoothness-Increasing Accuracy-Conserving Dirac-Delta Polynomial Kernels

Cited by 10 publications

References 37 publications

Method of Distributions for Systems with Stochastic Forcing

Method of Distributions for Systems with Stochastic Forcing

Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods

Uncertainty quantification in Eulerian-Lagrangian simulations of (point-)particle-laden flows with data-driven and empirical forcing models

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