Failsafe flux limiting and constrained data projections for equations of gas dynamics

Kuzmin, Dmitri; Möller, Matthias; Shadid, John N.; Shashkov, Mikhail

doi:10.1016/j.jcp.2010.08.009

Cited by 38 publications

(37 citation statements)

References 25 publications

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“…Variational formulations which lead to algebraic systems for calculating the coefficients U e i in continuous and discontinuous Galerkin (DG) methods can be found in [13,14,19] and [11,12], respectively. The main focus of this work is not on the way in which the Galerkin shape functions U e h are calculated but on postprocessing techniques designed to remove oscillatory antidiffusive components and enforce local maximum principles for selected variables.…”

Section: Discretization Of the Euler Equationsmentioning

confidence: 99%

“…Characteristic boundary conditions are imposed weakly using internal traces and freestream values in approximate Riemann solvers for the boundary terms. For a detailed description of the employed CG and DG discretizations, we refer the reader to [11,12,13,14,19].…”

Section: Numerical Examplesmentioning

confidence: 99%

“…We conclude that limiting techniques of this kind are well suited for shock capturing in finite element methods for computational gas dynamics. Numerical studies of synchronized positivity-preserving limiters for DG and CG approximations to the double Mach reflection problem were performed in [12,14,19]. Since no reference solution is available for this benchmark problem, we abstain from a quantitative comparison of the results produced by different limiters.…”

Section: Double Mach Reflectionmentioning

confidence: 99%

“…Whereas positivity preservation is a necessary condition for obtaining physically realistic solutions, it does not rule out the presence of undershoots and/or overshoots in the pressure distribution. On the other hand, limiting strategies that enforce local maximum principles for derived quantities of interest [14,23] frequently rely on linearized transformations of variables and cannot guarantee positivity preservation for the pressure and internal energy. This deficiency can be cured by means of a posteriori corrections, as proposed in [6,14,29].…”

Section: Introductionmentioning

confidence: 99%

“…Synchronized limiters [12,14,15,19,21] use the same correction factor for all components or a set of balanced correction factors for different components. The latter approach typically leads to constrained optimization problems [16] rather than closed-form expressions.…”

Section: Introductionmentioning

confidence: 99%

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Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations

Dobrev

Kolev

Kuzmin

et al. 2018

Journal of Computational Physics

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We present a new approach to enforcing local maximum principles in finite element schemes for the compressible Euler equations. In contrast to synchronized limiting techniques for systems of conservation laws, the density, momentum, and total energy are constrained in a sequential manner which guarantees positivity preservation for the pressure and internal energy. After the density limiting step, the total energy and momentum are adjusted to incorporate the irreversible effect of density changes. Then the corresponding antidiffusive corrections are limited to satisfy inequality constraints for the total and kinetic energy. The same element-based limiting strategy is employed in the context of continuous and discontinuous Galerkin methods. The sequential nature of the new limiting procedure makes it possible to achieve crisp resolution of contact discontinuities while using sharp local bounds in the energy constraints. A numerical study is performed for piecewise-linear finite element discretizations of 1D and 2D test problems.

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Section: Discretization Of the Euler Equationsmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Double Mach Reflectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations

Dobrev

Kolev

Kuzmin

et al. 2018

Journal of Computational Physics

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Implicit finite element schemes for the stationary compressible Euler equations

Gurris

Kuzmin

Turek

2011

Numerical Methods in Fluids

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SUMMARY A semi‐implicit finite element scheme and a Newton‐like solver are developed for the stationary compressible Euler equations. Since the Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable, the troublesome antidiffusive part is constrained within the framework of algebraic flux correction. A generalization of total variation diminishing (TVD) schemes is employed to blend the original Galerkin scheme with its nonoscillatory low‐order counterpart. Unlike standard TVD limiters, the proposed limiting strategy is fully multidimensional and readily applicable to unstructured meshes. However, the nonlinearity and nondifferentiability of the limiter function makes efficient computation of stationary solutions a highly challenging task, especially in situations when the Mach number is large in some subdomains and small in other subdomains. In this paper, a semi‐implicit scheme is derived via a time‐lagged linearization of the Jacobian operator, and a Newton‐like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. A boundary Riemann solver is used to avoid unphysical boundary states. It is shown that the proposed approach offers unconditional stability, as well as higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense. The overall spatial accuracy of the constrained scheme and the benefits of the new boundary treatment are illustrated by grid convergence studies for 2D benchmark problems. Copyright © 2011 John Wiley & Sons, Ltd.

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A shallow water event‐driven approach to simulate turbidity currents at stratigraphic scale

Santos

Lopes²,

Coutinho

2020

Numerical Methods in Fluids

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Summary We present a new event‐driven approach that combines a shallow water flow model with a practical sedimentation technique to simulate the formation of turbidite depositional systems at a stratigraphic scale. The equations that govern turbidity currents dynamics are solved using a new finite element flux‐corrected transport scheme. In this sense, the low‐order formulation is built by adding a novel Rusanov‐like scalar dissipation scaled by a shock‐capturing operator to standard Galerkin equations. From it, the high‐order system is obtained by including antidiffusive fluxes linearized around the low‐order solution and limited with the Zalesak's algorithm, following a minmod prelimiter. Implicit time integration with adaptive time steps is performed with an iterative nonlinear scheme that linearizes source terms. Sedimentation is implemented by carrying five granulometric fractions (clay, silt, and fine, medium, and coarse sands) along evolved streaklines and radially scattering sediments that deposit filling the available depositional space and compacting the underneath sediment layers. The flow is computed while an event discharge into an area of interest is active, or the inflow current has not reached an equilibrium state. Afterward, the event deposition step is executed. Numerical results of our flow solver presented a good agreement with available exact and literature solutions, and the simulated sediment deposits suggest that our approach is well suited for stratigraphic scale simulations.

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Failsafe flux limiting and constrained data projections for equations of gas dynamics

Cited by 38 publications

References 25 publications

Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations

Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations

Implicit finite element schemes for the stationary compressible Euler equations

A shallow water event‐driven approach to simulate turbidity currents at stratigraphic scale

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