Application of Relaxation Scheme to Degenerate Variational Inequalities

Babusikova, Jela

doi:10.1023/a:1013712628500

Cited by 3 publications

(4 citation statements)

References 13 publications

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“…Another possibility is to use linearisation by a relaxation scheme that was developed in [13,14] and has been justified in [1,2,11,21], etc. (see [16] for more details).…”

Section: Solution Of the Nonlinear Diffusion Problemmentioning

confidence: 99%

Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D

Frolkovič¹,

Kačur

2006

Comput Geosci

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A new method to determine semi-analytical solutions of one-dimensional contaminant transport problem with nonlinear sorption is described. This method is based on operator splitting approach where the convective transport is solved exactly and the diffusive transport by finite volume method. The exact solutions for all sorption isotherms of Freundlich and Langmuir type are presented for the case of piecewise constant initial profile and zero diffusion. Very precise numerical results for transport with small diffusion can be obtained even for larger time steps (e.g., when the Courant-Friedrichs-Lewy (CFL) condition failed). Mathematical modelThe main goal of this paper is to construct precise semi-analytical solutions of the nonlinear convectiondiffusion problem with adsorptionand boundary conditionsHere,where one can assume that δ = 1. The mathematical models (1)-(3) can represent contaminant transport in equilibrium mode with the sorption isotherm (u) = F(u) − u (see, e.g., [9]). The corresponding mathematical model isMost common forms of (nonlinear) sorption isotherms are Freundlichand Langmuirbut one can also consider isotherms of mixed type280 Comput Geosci (2006) 10: 279-290Equation (1) can be written in the form,where the function F (u) ≥ 1 can be viewed as a (nonlinear) "retardation" factor of the convective and diffusive transport.In the case of F (0) = ∞ [i.e., for (4) and (6) with 0 < p < 1], the solution of (1) can have a sharp front with a finite speed of propagation [27]. Moreover, the convective term can be dominant and thus the creation of shocks can be expected even for smooth initial data. Nevertheless, because of the presence of (small) diffusion D 0 > 0, the solution of (1)-(3) is regular and sharp shocks (as known for hyperbolic problems) can not develop in a finite time -see, e.g., [15].Precise numerical solutions of (1) are required, for instance, if one wishes to solve inverse problems (e.g., to determine the diffusion D and the sorption isotherm) or to test numerical methods for this type of problems. Most numerical methods so far are based on regularisation and/or upwinding procedures and they can produce undesirable artefacts in numerical solutions. The main goal of this paper is to avoid any regularisation of F(u) and to significantly decrease numerical dispersion. Consequently, precise numerical solutions of (1)-(3) can be obtained even for the case of vanishing diffusion and for very large retardation factor F (u).Characteristics-based numerical discretisations of contaminant transport with nonlinear adsorption were described in [3,4,8,15], and upwind-based discretization methods for the same type of problems in [6,22]. Asymptotic formulas for large-time behaviour of the exact solution of (1) were obtained in [7, 10], but they can not be used directly in discretization methods.The method presented in this paper is based on operator splitting, where nonlinear transport and nonlinear diffusion are solved separately along each time stepsee, e.g., [12]. The transport part is solved exactl...

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“…Another possibility is to use linearisation by a relaxation scheme that was developed in [13,14] and has been justified in [1,2,11,21], etc. (see [16] for more details).…”

Section: Solution Of the Nonlinear Diffusion Problemmentioning

confidence: 99%

Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D

Frolkovič¹,

Kačur

2006

Comput Geosci

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“…In the case of Dirichlet boundary conditions this is the resulting system of ODE, where we use C 0 (t) ≡ φ(t) (see (2), case a)). In the case of a Robin condition (see (2), case b)), the system ( 20)-( 21) has to by completed by the additional ODE at the point y = 0. In this case we make use of a ghost point y −1 = −α 1 .…”

Section: Numerical Approximation Of (1)mentioning

confidence: 99%

Numerical modelling of convection-diffusion-reaction problems with free boundary in 1D

Kacurova

2009

Preprint

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We discuss a numerical method for convection-diffusion-reaction problems with a free boundary in 1D. The method is based on the numerical modelling of the interface evolution, the transformation to a fixed domain problem and the approximation by an ODE system. The interface evolution is modelled by means of the local shape of the corresponding travelling wave solution. The method can be applied to many free boundary problems with a finite speed of the interface. The presented method can also approximate some problems with an infinite speed of the interface for damped travelling wave type solutions. In the numerical experiments we compare our numerical solution with the analytical ones for some problems.

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“…Richard's equation (3) has been solved (by finite volume approximations) in [12] under the assumption that b is Lipschitz continuous. The variational inequality (1) has been solved by the relaxation method in [3] without convective term and with a Lipschitz continuous b.…”

Section: (∂ T B(u) V −U)+(a∇u ∇(V −U))+(divf (X U) V −U)+(g(t U)mentioning

confidence: 99%

“…The approximation method is based on the relaxation method (to control the degeneracy of b) -see [3,15,[17][18][19]25], and on a modification of the method of characteristics -see [4, 5, 7-11, 20, 22, 28, 29] among others. We follow the idea of regularized characteristics analysed in [20].…”

Section: (∂ T B(u) V −U)+(a∇u ∇(V −U))+(divf (X U) V −U)+(g(t U)mentioning

confidence: 99%

Solution of degenerate parabolic variational inequalities with convection

Kačur¹,

Keer²

2003

ESAIM: M2AN

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Abstract. Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard's equation, modelling the unsaturated -saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.Mathematics Subject Classification. 65M25, 65M12.

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Application of Relaxation Scheme to Degenerate Variational Inequalities

Cited by 3 publications

References 13 publications

Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D

Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D

Numerical modelling of convection-diffusion-reaction problems with free boundary in 1D

Solution of degenerate parabolic variational inequalities with convection

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