A maximum-principle preserving finite element method for scalar conservation equations

Guermond, Jean‐Luc; Nazarov, Murtazo

doi:10.1016/j.cma.2013.12.015

Cited by 47 publications

(57 citation statements)

References 29 publications

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“…The piecewise-constant initial data is given by The inflow boundary conditions are defined using the exact solution of the pure initial value problem in R 2 . This solution can be found in [13] and stays in the invariant set G = [−1.0, 0.8].…”

Section: Inviscid Burgers Equationmentioning

confidence: 99%

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Kuzmin

2020

Computer Methods in Applied Mechanics and Engineering

113

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In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property-preserving low-order scheme. The addition of a second-order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions. Our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains, validity of local maximum principles, and entropy stability. The new methodology combines the advantages of modern entropy stable / entropy conservative schemes and their local extremum diminishing counterparts. The process of algebraic flux correction is based on inequality constraints which provably provide the desired properties. No free parameters are involved. The proposed algebraic fixes are readily applicable to unstructured meshes, finite element methods, general time discretizations, and steady-state residuals. Numerical studies of explicit entropy-constrained schemes are performed for linear and nonlinear test problems.

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Section: Inviscid Burgers Equationmentioning

confidence: 99%

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Kuzmin

2020

Computer Methods in Applied Mechanics and Engineering

113

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“…For arbitrary symmetric meshes the methods only differ on the weights of the terms in the sums in (13) and all the required properties stated in (22) are readily satisfied for the use of the shock detector in (21). In general meshes, the shock detectors are different, and the one in (21) is not linearity preserving.…”

Section: Nonlinear Stabilizationmentioning

confidence: 99%

Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization

Badia

Bonilla

2017

Computer Methods in Applied Mechanics and Engineering

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In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity).Peer ReviewedPostprint (author's final draft

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“…The inflow boundary conditions are defined using the exact solution of the pure initial value problem in R 2 . This solution can be found in [16] and stays in the invariant set G = [−1.0, 0.8]. The numerical solutions obtained at T = 0.5 using Q 2 elements with N h = 129 2 and N h = 257 2 DoFs are shown in Fig.…”

Section: Burgers Equationmentioning

confidence: 99%

Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws

Kuzmin

Luna

2020

Journal of Computational Physics

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This work extends the concepts of algebraic flux correction and convex limiting to continuous high-order Bernstein finite element discretizations of scalar hyperbolic problems. Using an array of adjustable diffusive fluxes, the standard Galerkin approximation is transformed into a nonlinear high-resolution scheme which has the compact sparsity pattern of the piecewise-linear or multilinear subcell discretization. The representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under CFL-like conditions. In contrast to predictor-corrector approaches based on the flux-corrected transport methodology, the proposed flux limiting strategy is monolithic; i.e., limited antidiffusive terms are incorporated into the well-defined residual of a nonlinear (semi-)discrete problem. A stabilized high-order Galerkin discretization is recovered if no limiting is performed. In the limited version, the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors. A formal proof of sparsity is provided for simplicial and box elements. The involved element contributions can be calculated efficiently making use of matrix-free algorithms and precomputed element matrices of the reference element. Numerical studies for Q 2 discretizations of linear and nonlinear two-dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions.

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A maximum-principle preserving finite element method for scalar conservation equations

Cited by 47 publications

References 29 publications

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization

Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws

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