Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

Mabuza, Sibusiso; Shadid, John N.; Kuzmin, Dmitri

doi:10.1016/j.jcp.2018.01.048

Cited by 14 publications

(20 citation statements)

References 33 publications

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“…Unfortunately, to the best of our knowledge, satisfying this definition does not ensure positivity of density, internal energy, or non-decreasing entropy. In any case, numerical schemes based on this definition have shown good numerical behavior [34,37,42,47].…”

Section: Definition 23 (Led)mentioning

confidence: 99%

“…Unfortunately, these ideas cannot be easily extended to implicit time integration and we are not aware of any implicit method that theoretically satisfies such properties. Kuzmin and co-workers [36,41,46,47] have proposed various schemes based on flux corrected transport (FCT) [44] that are experimentally robust, but lack of a theoretical analysis. Besides, this strategy also yields very stiff nonlinear problems.…”

Section: Introductionmentioning

confidence: 99%

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Monotonicity-preserving finite element schemes with adaptive mesh refinement for hyperbolic problems

Bonilla,

Badia

2019

Preprint

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This work is focused on the extension and assessment of the monotonicity-preserving scheme in [3] and the local bounds preserving scheme in [5] to hierarchical octree adaptive mesh refinement (AMR). Whereas the former can readily be used on this kind of meshes, the latter requires some modifications. A key question that we want to answer in this work is whether to move from a linear to a nonlinear stabilization mechanism pays the price when combined with shock-adapted meshes. Whereas nonlinear (or shock-capturing) stabilization leads to improved accuracy compared to linear schemes, it also negatively hinders nonlinear convergence, increasing computational cost. We compare linear and nonlinear schemes in terms of the required computational time versus accuracy for several steady benchmark problems. Numerical results indicate that, in general, nonlinear schemes can be cost-effective for sufficiently refined meshes. Besides, it is also observed that it is better to refine further around shocks rather than use sharper shock capturing terms, which usually yield stiffer nonlinear problems. In addition, a new refinement criterion has been proposed. The proposed criterion is based on the graph Laplacian used in the definition of the stabilization method. Numerical results show that this shock detector performs better than the well-known Kelly estimator for problems with shocks or discontinuities.

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Section: Definition 23 (Led)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Monotonicity-preserving finite element schemes with adaptive mesh refinement for hyperbolic problems

Bonilla,

Badia

2019

Preprint

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“…Representatives of this class of correction factors were constructed and successfully applied, e.g., in [BB17;Bar+17b;Bar+17a;Kuz+17]. Extensions of the AFC methodology to nonlinear systems of hyperbolic problems can found, e.g., in [Kuz07;Bad+19;Mab+18;Kuz20].…”

Section: Introductionmentioning

confidence: 99%

An algebraic flux correction scheme facilitating the use of Newton-like solution strategies

Lohmann

2021

Computers & Mathematics with Applications

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Building on recent advances in the analysis and design of algebraic flux correction (AFC) schemes, new differentiable limiter functions are constructed for efficient nonlinear solution strategies. The proposed scaling parameters are used to limit artificial diffusion operators incorporated into the residual of a high order target scheme to produce accurate and boundpreserving finite element approximations to hyperbolic problems. Due to this stabilization procedure, the occurring system becomes highly nonlinear and the efficient computation of corresponding solutions is a challenging task. The presented regularization approach makes the AFC residual twice continuously differentiable so that Newton's method converges quadratically for sufficiently good initial guesses. Furthermore, the performance of each nonlinear iteration is improved by expressing the Jacobian as the sum and product of matrices having the same sparsity pattern as the Galerkin system matrix. Eventually, the AFC methodology constructed for stationary problems is extended to transient test cases and validated numerically by applying it to several numerical benchmarks.

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“…Hence, the stabilization methods are based on adapting the scalar techniques for characteristic variables to the system written in the original set of variables. Following this strategy, some progress has been recently made in stabilized FE schemes by making use of flux corrected transport (FCT) algorithms [22,26,28,29]. The schemes proposed therein are based in two main ingredients.…”

mentioning

confidence: 99%

“…In the current work, we extend the differentiable nonlinear stabilization in [2] to the Euler equations using the ideas from [22,29] to define the artificial diffusion operators for hyperbolic systems of equations. The new method is applied to the steady and transient Euler equations, and its nonlinear convergence is assessed.…”

mentioning

confidence: 99%

On differentiable local bounds preserving stabilization for Euler equations

Badia,

Bonilla,

Mabuza

et al. 2019

Preprint

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This work is focused on the design of nonlinear stabilization techniques for the finite element approximation of the Euler equations in steady form and the implicit time integration of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. We attempt to mitigate this issue by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. The importance of the differentiability is assessed by comparing the developed scheme with its nondifferentiable counterpart. Numerical experiments suggest that, up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. In the case of transient problem, we also observe a reduction in the computational cost.

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Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

Cited by 14 publications

References 33 publications

Monotonicity-preserving finite element schemes with adaptive mesh refinement for hyperbolic problems

Monotonicity-preserving finite element schemes with adaptive mesh refinement for hyperbolic problems

An algebraic flux correction scheme facilitating the use of Newton-like solution strategies

On differentiable local bounds preserving stabilization for Euler equations

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