Shock-layer bounds for a singularly perturbed equation

Scroggs, Jeffrey S.

doi:10.1090/qam/1343460

Cited by 2 publications

(6 citation statements)

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“…The numerical examples given here indicate that the correction (antidiffusive) effect is significant in the shock layer regions when the residual flux term is used in an explicit correction step (25). The front tracking method plays an essential role in the construction of the residual flux term (27).…”

Section: Discussionmentioning

confidence: 95%

“…Our only interest is to see if COS can resolve the shock layer using one time step. For the problems under consideration we know that the size of the shock layer should be O(ε); see [27]. We see that the layer produced by OS is (several) orders of magnitude too wide.…”

Section: A Fully Discrete Methodmentioning

confidence: 93%

“…This can be inferred with the fact that the entropy condition (Oleinik's convexification criterion [28]) forces the hyperbolic solver to throw away information (entropy loss) regarding the structure of nonlinear shock fronts. We have shown that it is possible to construct a residual flux term (27) which can be employed in a third (correction) step to counterbalance the entropy loss. The purpose of the correction step is to ensure correct structure of nonlinear shock fronts.…”

Section: Discussionmentioning

confidence: 99%

“…Remark 1. According to (27), a local residual term f i res is constructed with respect to each discontinuity in the piecewise constant front tracking solution. In particular, a large number of these local flux terms will be so small that they have no significant influence on the solution.…”

Section: The Correction Operatormentioning

confidence: 99%

“…Let us elaborate on this feature (splitting error) by studying an application of OS to Burgers's equation [3], i.e., f (u) = In particular, the size of the shock layer is O(ε) (see, e.g., [27]), which contrasts with the well-known O( √ ε)-layers seen in linear equations. Let S f (t) denote the entropy satisfying solution operator associated with the nonlinear conservation law…”

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confidence: 99%

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Corrected Operator Splitting for Nonlinear Parabolic Equations

Karlsen¹,

Risebro²

2000

SIAM J. Numer. Anal.

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Abstract. We present a corrected operator splitting (COS) method for solving nonlinear parabolic equations of a convection-diffusion type. The main feature of this method is the ability to correctly resolve nonlinear shock fronts for large time steps, as opposed to a standard operator splitting (OS) which fails to do so. COS is based on solving a conservation law for modeling convection, a heat-type equation for modeling diffusion and finally a certain "residual" conservation law for necessary correction. The residual equation represents the entropy loss generated in the hyperbolic (convection) step. In OS the entropy loss manifests itself in the form of too wide shock fronts. The purpose of the correction step in COS is to counterbalance the entropy loss so that correct width of nonlinear shock fronts is ensured. The polygonal method of Dafermos [J. Math. Anal. Appl., 38 (1972), pp. 33-41] constitutes an important part of our solution strategy. It is shown that COS generates a compact sequence of approximate solutions which converges to the solution of the problem. Finally, some numerical examples are presented where we compare OS and COS methods with respect to accuracy.

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Section: Discussionmentioning

confidence: 95%

Section: A Fully Discrete Methodmentioning

confidence: 93%

Section: Discussionmentioning

confidence: 99%

Section: The Correction Operatormentioning

confidence: 99%

mentioning

confidence: 99%

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Corrected Operator Splitting for Nonlinear Parabolic Equations

Karlsen¹,

Risebro²

2000

SIAM J. Numer. Anal.

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The corrected operator splitting approach applied to a nonlinear advection-diffusion problem

Karlsen

Brusdal

Dahle

et al. 1998

Computer Methods in Applied Mechanics and Engineering

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So-called corrected operator splitting methods are applied to a 1-D scalar advection diffusion equation of Buckley-Leverett type with general initial data. Front tracking and a 2nd order Godunov method are used to advance the solution in time. Diffusion is modelled by piecewise linear finite elements at each new time level. To obtain correct structure of shock fronts independently of the size of the time step, a dynamically defined residual flux term is grouped with diffusion. Different test problems are considered, and the methods are compared with respect to accuracy and runtime. Finally, wc extend the corrected operator splitting to 2-D equations by means of dimensional splitting, and wc apply it to a Buckley-Leverett problem including gravitational effects. KARLSEN, BRUSDAL, DAHLE, EVJE, AND LIE simple analytic expressionThe capillary diffuse coefficient u(u) is generally a nonlinear (bell-shaped) funct lOn of u, which we recreate using the expression We note that the diffusion coefficient becomes zero at u = 0, 1, so that (1) is an example of a degenerate parabolic equation; see Wpert and Hudjaev [37] for global existence, uniqueness, and stability results in BV space for the initial value problem with general "(") > 0. and Zhuo-qun and Jun-yu [42] for similar results concerning the initial-boundary value problem. Properties of BV solutions, such as regularity, have been studied by Jun-Ning [25]. For further results on degenerate equations we refer to a recent survey paper by Chen [6] and the references therein.The scaling parameter e in front of the capillary diffusion term is usually small for reasonable flow rates. Consequently, this term can be neglected if the main problem is to trace fluid interfaces in which case the problem is reduced to solving a hyperbolk conservation law. However, in many applications some detailed information on the structure of fronts is important, and the diffusion term cannot be neglected. Since we believe that it is important to obtain correct placement and

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Shock-layer bounds for a singularly perturbed equation

Cited by 2 publications

References 6 publications

Corrected Operator Splitting for Nonlinear Parabolic Equations

Corrected Operator Splitting for Nonlinear Parabolic Equations

The corrected operator splitting approach applied to a nonlinear advection-diffusion problem

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