Numerical Methods for Linear Equations

LeVeque, Randall J.

doi:10.1007/978-3-0348-8629-1_10

Cited by 4 publications

(4 citation statements)

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“…We provide two examples of problems of the form (2.1) at the end of this section. We remark that the solution may contain discontinuities and might not be unique: we here seek satisfying (2.1) away from discontinuities and satisfying suitable Rankine-Hugoniot and entropy conditions at discontinuities, [27]. We recast (2.1) as ∇ • ( ) = ( ) in Ω (2.2)…”

Section: Space-time Formulation Of Conservation Lawsmentioning

confidence: 99%

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

Taddei

Zhang

2021

ESAIM: M2AN

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We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov - Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We present numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.

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Section: Space-time Formulation Of Conservation Lawsmentioning

confidence: 99%

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

Taddei

Zhang

2021

ESAIM: M2AN

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“…By the standard theory of hyperbolic conservation laws (14), the RP (with concave flux) has the following unique entropy solution:…”

Section: Riemann Problem and Cell Transmission Modelmentioning

confidence: 99%

Off-Ramp Coupling Conditions Devoid of Spurious Blocking and Re-Routing

Salehi

Somers

Seibold

2018

Transportation Research Record

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When modeling vehicular traffic flow on highway networks via macroscopic models, suitable coupling conditions at the network nodes are crucial. Frequently, the evolution of traffic flow on each network edge is described in a lane-averaged fashion using a singleclass Lighthill-Whitham-Richards model. At off-ramps, split ratios (i.e., what percentage of traffic exits the highway) are prescribed that can be drawn from historic data. In this situation, classical FIFO coupling conditions yield unrealistic results, in that a clogged off-ramp yields zero flux through the node. As a remedy, non-FIFO conditions have been proposed. However, as we demonstrate here, those lead to spurious re-routing of vehicles. Then, a new coupling model, FIFOQ, is presented that preserves the desirable properties of non-FIFO models, while not leading to any spurious re-routing.2000 Mathematics Subject Classification. 35L65; 35Q91; 91B74.

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“…, J − 1, or until density approximations spanning the duration of the simulation have been generated. In any such explicit scheme, it is essential, for stability, to ensure that the Courant-Friedrichs-Lewy condition [see LeVeque (17 )] that cell size not be less than the product of time step and local wave speed not be violated too egregiously.…”

Section: Conservation Of Vehicles Then Givesmentioning

confidence: 99%

“…("Entropy" per se is not a concept that appears not to have been defined in the context of traffic flow. For scalar conservation laws such as the KWM, various forms of an "entropy condition" have been defined [e.g., see section 3.8 of LeVeque (17 )] to select the relevant (weak) solution from the multiplicity of solutions that otherwise would occur. That solution is known as the "entropy solution."…”

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confidence: 99%

Point Constriction, Interface, and Boundary Conditions for Kinematic-Wave Model

Nelson

Kumar

2006

Transportation Research Record

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The entropy solutions of the Riemann problem for the kinematic-wave model are described for (entrant and exiting) boundary conditions, interfaces (e.g., lane drops), and point constrictions (e.g., bottlenecks or signalized intersections) for kinematic-wave models. These results are used to obtain analytic solutions and computational solutions, by the method of Godunov, for the three test scenarios earlier suggested by Ross. Results show that the kinematic-wave model, with a correct computational solution (i.e., converging to the entropy solution), correctly produces the analytic (entropy) solutions of the kinematic-wave model and that analytic and computational solutions both satisfy intuitive expectations at least as well as the novel model suggested by Ross and even avoid some acknowledged deficiencies of the latter. They also emphasize the necessity of using discrete approximations that satisfy the entropy condition and of exercising care in ensuring that the discretization parameters are sufficiently small.

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Numerical Methods for Linear Equations

Cited by 4 publications

References 0 publications

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

Off-Ramp Coupling Conditions Devoid of Spurious Blocking and Re-Routing

Point Constriction, Interface, and Boundary Conditions for Kinematic-Wave Model

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