On differentiable local bounds preserving stabilization for Euler equations

Badia, Santiago; Bonilla, Jesús; Mabuza, Sibusiso; Shadid, John N.

doi:10.1016/j.cma.2020.113267

Cited by 6 publications

(11 citation statements)

References 24 publications

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“…In this section, we describe the additional terms used for the stabilized problem (2). In particular, we use the stabilization terms defined in [3] for the scalar problem and [5] for Euler:…”

Section: Nonlinear Stabilizationmentioning

confidence: 99%

“…Besides, this strategy also yields very stiff nonlinear problems. Differentiable schemes for compressible flows have been proposed in [5] to alleviate (but not eliminate) this problem.…”

Section: Introductionmentioning

confidence: 99%

“…The motivation of this work is to shed light on this issue. First, we adapt the schemes developed in [3,5] to hierarchical octree AMR [9,54]. Next, we propose a refinement criterium that relies on information already present in the stabilization technique; nonlinear stabilization methods include a shock detector to activate the artificial diffusion only close to discontinuities.…”

Section: Introductionmentioning

confidence: 99%

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mentioning

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: Nonlinear Stabilizationmentioning

confidence: 99%

“…Besides, this strategy also yields very stiff nonlinear problems. Differentiable schemes for compressible flows have been proposed in [5] to alleviate (but not eliminate) this problem.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

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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“…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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Evaluation of an effective and robust implicit time-integration numerical scheme for Navier-Stokes equations in a CFD solver for compressible flows

Maia

Cavalca

Tomita

et al. 2022

Applied Mathematics and Computation

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On differentiable local bounds preserving stabilization for Euler equations

Cited by 6 publications

References 24 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

Evaluation of an effective and robust implicit time-integration numerical scheme for Navier-Stokes equations in a CFD solver for compressible flows

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