Smooth and Compactly Supported Viscous Sub-cell Shock Capturing for Discontinuous Galerkin Methods

Glaubitz, Jan; Nogueira, Alberto Costa; Almeida, J.L.S.; Cantão, Renato Fernandes; Silva, C.A.C.

doi:10.1007/s10915-018-0850-3

Cited by 20 publications

(14 citation statements)

References 72 publications

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“…One of the main challenges in solving nonlinear conservation laws (2) is balancing high resolution properties and the amount of viscosity introduced to maintain stability, especially near shocks [5,7,11]. Applying the techniques presented in §2.2 and §2.3, we are now able to adapt the nodal DG method described in §2.1 to include 1 regularization.…”

Section: Methodsmentioning

confidence: 99%

See 1 more Smart Citation

High Order Edge Sensors with $\ell^1$ Regularization for Enhanced Discontinuous Galerkin Methods

Glaubitz

Gelb

2019

SIAM J. Sci. Comput.

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This paper investigates the use of 1 regularization for solving hyperbolic conservation laws based on high order discontinuous Galerkin (DG) approximations. We first use the polynomial annihilation method to construct a high order edge sensor which enables us to flag "troubled" elements. The DG approximation is enhanced in these troubled regions by activating 1 regularization to promote sparsity in the corresponding jump function of the numerical solution. The resulting 1 optimization problem is efficiently implemented using the alternating direction method of multipliers. By enacting 1 regularization only in troubled cells, our method remains accurate and efficient, as no additional regularization or expensive iterative procedures are needed in smooth regions. We present results for the inviscid Burgers' equation as well as for a nonlinear system of conservation laws using a nodal collocation-type DG method as a solver.

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Section: Methodsmentioning

confidence: 99%

“…1 At least for smooth solutions, discontinuous Galerkin methods are capable of spectral orders of accuracy. 1 regularization as well as any other shock capturing procedure [5,7,10,11,13] should thus be just applied in (and near) elements where discontinuities are present. We refer to those elements as troubled elements.…”

Section: Discontinuity Sensormentioning

confidence: 99%

High Order Edge Sensors with $\ell^1$ Regularization for Enhanced Discontinuous Galerkin Methods

Glaubitz

Gelb

2019

SIAM J. Sci. Comput.

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“…Furthermore, it appears that the differentiation matrices of RBF methods encountered in time-dependent PDEs often have eigenvalues with a positive real part resulting in unstable methods; see [76]. Hence, in the presence of rounding errors, these methods are less accurate [55,75,80] and can become unstable in time unless a dissipative time integration method [64,76], artificial dissipation [22,77,39,37,73], or some other stabilizing technique [79,28,35,48,40,30,15] is used. So far, this issue was only overcome for problems which are free of BCs [64].…”

Section: State Of the Artmentioning

confidence: 99%

The physics of financial networks

et al. 2021

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As the total value of the global financial market outgrew the value of the real economy, financial institutions created a global web of interactions that embodies systemic risks. Understanding these networks requires new theoretical approaches and new tools for quantitative analysis. Statistical physics contributed significantly to this challenge by developing new metrics and models for the study of financial network structure, dynamics, and stability and instability. In this Review, we introduce network representations originating from different financial relationships, including direct interactions such as loans, similarities such as co-ownership and higher-order relations such as contracts involving several parties (for example, credit default swaps) or multilayer connections (possibly extending to the real economy). We then review models of financial contagion capturing the diffusion and impact of shocks across each of these systems. We also discuss different notions of 'equilibrium' in economics and statistical physics, and how they lead to maximum entropy ensembles of graphs, providing tools for financial network inference and the identification of early-warning signals of system-wide instabilities.

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“…This idea dates back to the pioneering work [57] of von Neumann and Richtmyer during the Manhattan project in the 1940's at Los Almos National Laboratory, where they constructed stable FD schemes for the equations of hydrodynamics by including artificial viscosity (AV) terms. Since then, AV methods have attracted the interest of many researchers and were investigated also for SE approximations in a large number of works [18,19,31,43,48]. Despite providing a robust and accurate way to capture (shock) discontinuities, AV terms are not trivial to include in SE approximations.…”

Section: Introductionmentioning

confidence: 99%

“…Typically, they are nonlinear and consist of higher (second and fourth order) derivatives. Another drawback arises from the fact that AV terms can introduce additional harsh time step restrictions, when not constructed with care, and thus decrease the efficiency of the numerical method [18,25]. Finally, we mention those methods based on order reduction [5,10], mesh adaptation [14], weighted essentially nonoscillatory (WENO) concepts [52,53], and 1 regularisation applied to high order approximations of the jump function [17].…”

Section: Introductionmentioning

confidence: 99%

Shock capturing by Bernstein polynomials for scalar conservation laws

Glaubitz

2019

Applied Mathematics and Computation

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A main disadvantage of many high-order methods for hyperbolic conservation laws lies in the famous Gibbs-Wilbraham phenomenon, once discontinuities appear in the solution. Due to the Gibbs-Wilbraham phenomenon, the numerical approximation will be polluted by spurious oscillations, which produce unphysical numerical solutions and might finally blow up the computation. In this work, we propose a new shock capturing procedure to stabilise high-order spectral element approximations. The procedure consists of going over from the original (polluted) approximation to a convex combination of the original approximation and its Bernstein reconstruction, yielding a stabilised approximation. The coefficient in the convex combination, and therefore the procedure, is steered by a discontinuity sensor and is only activated in troubled elements. Building up on classical Bernstein operators, we are thus able to prove that the resulting Bernstein procedure is total variation diminishing and preserves monotone (shock) profiles. Further, the procedure can be modified to not just preserve but also to enforce certain bounds for the solution, such as positivity. In contrast to other shock capturing methods, e. g. artificial viscosity methods, the new procedure does not reduce the time step or CFL condition and can be easily and efficiently implemented into any existing code. Numerical tests demonstrate that the proposed shock-capturing procedure is able to stabilise and enhance spectral element approximations in the presence of shocks.

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Smooth and Compactly Supported Viscous Sub-cell Shock Capturing for Discontinuous Galerkin Methods

Cited by 20 publications

References 72 publications

High Order Edge Sensors with $\ell^1$ Regularization for Enhanced Discontinuous Galerkin Methods

High Order Edge Sensors with $\ell^1$ Regularization for Enhanced Discontinuous Galerkin Methods

The physics of financial networks

Shock capturing by Bernstein polynomials for scalar conservation laws

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