Positivity Preserving Limiters for Time-Implicit Higher Order Accurate Discontinuous Galerkin Discretizations

J.J.W. van der Vegt; Yan Xu; Yinhua Xia

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Positivity Preserving Limiters for Time-Implicit Higher Order Accurate Discontinuous Galerkin Discretizations

J.J.W. van der Vegt, Yinhua Xia, Yan Xu

Numerical Analysis · Submitted 21 Nov 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The time-implicit positivity preserving DG discretization is demonstrated for several nonlinear scalar conservation laws, which include the advection, Burgers, Allen-Cahn, Barenblatt, and Buckley-Leverett equations.

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arXiv ID1811.08620 (version 3)

Pith IDFC74V2IV, Pith's permanent record of this paper and its review

CitationJ.J.W. van der Vegt, Yan Xu, Yinhua Xia. "Positivity Preserving Limiters for Time-Implicit Higher Order Accurate Discontinuous Galerkin Discretizations." arXiv:1811.08620 (2018). https://pith.science/paper/FC74V2IV

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What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.66 · human participants 0.02 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.61 · mentions code or data availability 0.02.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Currently, nearly all positivity preserving discontinuous Galerkin (DG) discretizations of partial differential equations are coupled with explicit time integration methods. Unfortunately, for many problems this can result in severe time-step restrictions. The techniques used to develop explicit positivity preserving DG discretizations can, however, not easily be combined with implicit time integration methods. In this paper we therefore present a new approach. Using Lagrange multipliers the conditions imposed by the positivity preserving limiters are directly coupled to a DG discretization combined with a Diagonally Implicit Runge-Kutta time integration method. The positivity preserving DG discretization is then reformulated as a Karush-Kuhn-Tucker (KKT) problem, which is frequently encountered in constrained optimization. Since the limiter is only active in areas where positivity must be enforced it does not affect the higher order DG discretization elsewhere. The resulting non-smooth nonlinear algebraic equations have, however, a different structure compared to most constrained optimization problems. We therefore develop an efficient active set semi-smooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Convergence of this semi-smooth Newton method is proven using a specially designed quasi-directional derivative of the time-implicit positivity preserving DG discretization. The time-implicit positivity preserving DG discretization is demonstrated for several nonlinear scalar conservation laws, which include the advection, Burgers, Allen-Cahn, Barenblatt, and Buckley-Leverett equations.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Currently, nearly all positivity preserving discontinuous Galerkin (DG) discretizations of partial differential equations are coupled with explicit time integration methods.	background 0.99	0.05	0.02	asserted 1.00
s1	Unfortunately, for many problems this can result in severe time-step restrictions.	background 0.92	0.09	0.02	asserted 0.72
s2	The techniques used to develop explicit positivity preserving DG discretizations can, however, not easily be combined with implicit time integration methods.	gap 0.65	0.10	0.03	asserted 0.85
s3	In this paper we therefore present a new approach.	contribution 0.99	0.41	0.04	asserted 0.75
s4	Using Lagrange multipliers the conditions imposed by the positivity preserving limiters are directly coupled to a DG discretization combined with a Diagonally Implicit Runge-Kutta time integration method.	method 0.89	0.92	0.14	asserted 1.00
s5	The positivity preserving DG discretization is then reformulated as a Karush-Kuhn-Tucker (KKT) problem, which is frequently encountered in constrained optimization.	method 0.94	0.88	0.08	asserted 1.00
s6	Since the limiter is only active in areas where positivity must be enforced it does not affect the higher order DG discretization elsewhere.	method 0.42	0.71	0.07	asserted 1.00
s7	The resulting non-smooth nonlinear algebraic equations have, however, a different structure compared to most constrained optimization problems.	gap 0.36	0.70	0.03	asserted 1.00
s8	We therefore develop an efficient active set semi-smooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations.	method 0.60	0.96	0.20	asserted 1.00
s9	Convergence of this semi-smooth Newton method is proven using a specially designed quasi-directional derivative of the time-implicit positivity preserving DG discretization.	contribution 0.70	0.93	0.13	asserted 1.00
s10	The time-implicit positivity preserving DG discretization is demonstrated for several nonlinear scalar conservation laws, which include the advection, Burgers, Allen-Cahn, Barenblatt, and Buckley-Leverett equations.	result 0.55	0.95	0.26	asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.67.

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Original abstract

Currently, nearly all positivity preserving discontinuous Galerkin (DG) discretizations of partial differential equations are coupled with explicit time integration methods. Unfortunately, for many problems this can result in severe time-step restrictions. The techniques used to develop explicit positivity preserving DG discretizations can, however, not easily be combined with implicit time integration methods. In this paper we therefore present a new approach. Using Lagrange multipliers the conditions imposed by the positivity preserving limiters are directly coupled to a DG discretization combined with a Diagonally Implicit Runge-Kutta time integration method. The positivity preserving DG discretization is then reformulated as a Karush-Kuhn-Tucker (KKT) problem, which is frequently encountered in constrained optimization. Since the limiter is only active in areas where positivity must be enforced it does not affect the higher order DG discretization elsewhere. The resulting non-smooth nonlinear algebraic equations have, however, a different structure compared to most constrained optimization problems. We therefore develop an efficient active set semi-smooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Convergence of this semi-smooth Newton method is proven using a specially designed quasi-directional derivative of the time-implicit positivity preserving DG discretization. The time-implicit positivity preserving DG discretization is demonstrated for several nonlinear scalar conservation laws, which include the advection, Burgers, Allen-Cahn, Barenblatt, and Buckley-Leverett equations.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites it as background 1 paper

Some of these methods modify the spatial discretization [ref] , and others use limiters derived from Karush-Kahan-Tucker (KKT) optimality conditions [ref] .

Structure-preserving Nonlinear Filtering for Continuous and Discontinuous Galerkin Spectral/hp Element Methods · Existing techniques for structure preservation · 0.96

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