Front Tracking for Hyperbolic Conservation Laws

Holden, Helge; Risebro, Nils Henrik

doi:10.1007/978-3-642-23911-3

Cited by 207 publications

(296 citation statements)

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“…Note that Godunov's flux has less regularity. For f ∈ C 1 , the spatial L 1 error is O(( z) r ) with r = 1/2; see [43][44][45]. Consider now the more general strongly degenerate parabolic equation X t + f (X) z = D(X ) zz , which is the type we have within the clarification and thickening zones.…”

Section: Efficiencies Of Numerical Methodsmentioning

confidence: 97%

Fast reliable simulations of secondary settling tanks in wastewater treatment with semi-implicit time discretization

Diehl

Fars

Mauritsson

2015

Computers & Mathematics with Applications

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a b s t r a c tThe bio-kinetic and sedimentation processes of wastewater treatment plants can be modelled by a large system of coupled nonlinear ordinary and partial differential equations (ODEs and PDEs). The subprocess of continuous sedimentation, which contains concentration discontinuities, is modelled by a degenerate parabolic conservation PDE with spatially discontinuous coefficients. A spatial discretization of this PDE described in Bürger et al. (2013) results in a large system of method-of-lines ODEs for the entire plant and simulation can be performed by integration in time. In practice, standard time integration methods available in commercial simulators are often used. Shortages of such methods are here shown, such as the smearing of shock waves by Runge-Kutta methods and long execution times. A semi-implicit time discretization, which is described in detail, provides substantially shorter computational times and is more efficient than standard methods.

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Section: Efficiencies Of Numerical Methodsmentioning

confidence: 97%

Fast reliable simulations of secondary settling tanks in wastewater treatment with semi-implicit time discretization

Diehl

Fars

Mauritsson

2015

Computers & Mathematics with Applications

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“…It consists in computing two compression fluxes K^\"j and K^l"-, using respectively the x-and y-derivatives in the equation (5). Then these fluxes are apphed alternatively in each direction, using nearly the same formula as (6), up to the CFL condition that gets a little more restrictive (X < j). As we will see in the numerical results presented in the talk, this version of ACM has inherited from the very good compressivity of its one-dimensional counterpart.…”

Section: Extension Of Acm To Cartesian Two-dimensional Gridsmentioning

confidence: 99%

“…We therefore will only be applying ACM, once again using (6) in the ^-direction. The determination ofK^ \"j will still follow (5), except that directional derivatives will be used.…”

Section: The Directional Approachmentioning

confidence: 99%

“…Some are based on limiters (e.g. [5]), some use random values, like Glimm's scheme, some use front tracking (see [6]), etc. Among them is Harten's ACM (Artificial Compression Method).…”

Section: Introductionmentioning

confidence: 99%

“….." X 1 (6) which is monotonicity preserving under the CFL condition (here A < 1) and will produce the best possible resolution of stationary shocks [4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Enhancing the Capture of Two-dimensional, Shock-Induced Detonation Fronts using Harten’s Artificial Compression Method on Underresolved Cartesian Grids

Rouch¹

2009

AIP Conference Proceedings

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In the problems of detonation we are going to present here, the capture of shocks is crucial, since from its accuracy depends the very physical relevance of the whole solution. We already obtained some improvements in the capture of detonation fronts on underresolved grids in the one-dimensional case thanks to Harten's artificial compression [1], [2], and we now turn to two-dimensional cartesian grids. In two space dimensions, the complexity of the discontinuity implies not only the location and speed of the front, but also its shape. These three aspects will be explored here, for the ZND detonation model, based on Euler's system. We will use two-dimensional central schemes for main computations, namely the classical "Lax-Friedrichs" first order scheme and the second order "Jiang-Tadmor" scheme [3]. To each of these methods, we will add two different versions of ACM (Artificial Compression Method), one that uses space splitting, and the other based on directional differencing [4]. Finally, as ACM require a good knowledge of the regions of the solution potentially carrying discontinuities, it will be assisted by a DoD (Detector of Discontinuities) based on the entropy production rate.

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Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

Adimurthi

Ghoshal

Dutta

et al. 2010

Comm Pure Appl Math

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For the scalar conservation laws with discontinuous flux, an infinite family of .A; B/-interface entropies are introduced and each one of them is shown to form an L 1 -contraction semigroup (see [2]). One of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution. Away from the interface, boundedness of total variation of the solution has been proved in a recent paper [6]. In this paper, we discuss this particular issue in detail and produce a counterexample to show that the solution, in general, has unbounded total variation near the interface. In fact, this example illustrates that smallness of the BV norm of the initial data is immaterial. We hereby settle the question of determining for which of the aforementioned .A; B/ pairs the solution will have bounded total variation in the case of strictly convex fluxes.

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Front Tracking for Hyperbolic Conservation Laws

Cited by 207 publications

References 0 publications

Fast reliable simulations of secondary settling tanks in wastewater treatment with semi-implicit time discretization

Fast reliable simulations of secondary settling tanks in wastewater treatment with semi-implicit time discretization

Enhancing the Capture of Two-dimensional, Shock-Induced Detonation Fronts using Harten’s Artificial Compression Method on Underresolved Cartesian Grids

Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

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