Approximation of Degenerate Parabolic Problems Using Numerical Integration

Nochetto, Ricardo H.; Verdi, C.

doi:10.1137/0725046

Cited by 148 publications

(163 citation statements)

References 32 publications

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“…We require, for physical coherence, that X (S) is strictly increasing for 0 < S < 1. By using (33) and (34),…”

Section: Mathematical Model When Capillary Effects Are Non-negligiblementioning

confidence: 99%

“…Concerning the situation with non-negligible capillary pressure, there are many references when the diffusive F vanishes after a threshold valued. We mention the reference [6,17,18,22,33,36,37,38,39]. In our approach we allow both degenerate parabolic and hyperbolic situations.…”

Section: And If It Satisfies the Following Entropy Inequalities For Amentioning

confidence: 99%

“…We use mass lumping approach [33]- [34] with weakly-acute mesh to garanty the monotonicity. For the diffusion part we use the implicit scheme and for the non-linear term in the time derivative the explicit scheme.…”

Section: Discrete Scheme Used In the Mmodd-scilab Implementationmentioning

confidence: 99%

See 2 more Smart Citations

Isothermal water flows in low porosity porous media in presence of vapor–liquid phase change

Farína¹,

Bodin

Clopeau

et al. 2014

Nonlinear Analysis: Real World Applications

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In this article we consider a mathematical model for a low porosity porous medium saturated by water, present both as the liquid and the vapor phase. In the isothermal case we propose a new formulation using a single nonlinear parabolic-hyperbolic equation for the fluid mixture density X. We present the derivation of the unified model and a number of numerical simulations based on regularization and Kirchhoff's transform.

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“…We require, for physical coherence, that X (S) is strictly increasing for 0 < S < 1. By using (33) and (34),…”

Section: Mathematical Model When Capillary Effects Are Non-negligiblementioning

confidence: 99%

Section: And If It Satisfies the Following Entropy Inequalities For Amentioning

confidence: 99%

See 1 more Smart Citation

Isothermal water flows in low porosity porous media in presence of vapor–liquid phase change

Farína¹,

Bodin

Clopeau

et al. 2014

Nonlinear Analysis: Real World Applications

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show abstract

“…We also mention two relevant works of Nochetto and Verdi [17] and Elliott [10], concerning the effect of numerical integration in multidimensional degenerate parabolic problems.…”

Section: Error Boundsmentioning

confidence: 99%

Error Estimates for a Nonlinear Degenerate Parabolic Problem

Lesaint¹,

Pousin²

1992

Mathematics of Computation

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Abstract.In this paper we are dealing with a partial differential equation of parabolic type, which degenerates on one side of the domain. This equation may be viewed either as a model of particle diffusion in plasma physics, or as a simplified model of a viscous boundary layer in two dimensions. Known results for the existence and uniqueness of the weak solution are first recalled. A finite difference implicit scheme is then defined, and error bounds are derived, taking into account the low degree of smoothness of the exact solution. An iterative algorithm for the computation of the numerical solution at each time step is shown to be convergent.

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“…The existence and uniqueness theory for the Stefan problem is well understood; however, the numerical analysis for such problems is crude and gives sub-optimal results, [17,8,18]. Part of the problem is the lack of regularity of the solution which may be summarized as follows.…”

mentioning

confidence: 99%

Co-volume methods for degenerate parabolic problems

Baughman

Walkington

1993

Numer. Math.

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Abstract. A complementary volume (co-volume) technique is used to develop a physically appealing algorithm for the solution of degenerate parabolic problems, such as the Stefan problem. It is shown that, these algorithms give rise to a discrete semigroup theory that parallels the continuous problem. In particular, the discrete Stefan problem gives rise to nonlinear semigroups in both the discrete L 1 and H~x spaces.

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Approximation of Degenerate Parabolic Problems Using Numerical Integration

Cited by 148 publications

References 32 publications

Isothermal water flows in low porosity porous media in presence of vapor–liquid phase change

Isothermal water flows in low porosity porous media in presence of vapor–liquid phase change

Error Estimates for a Nonlinear Degenerate Parabolic Problem

Co-volume methods for degenerate parabolic problems

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