Solution of Nonlinear Diffusion Problems by Linear Approximation Schemes

Kačur, Jozef; Handlovičová, Angela; Kačurová, M.

doi:10.1137/0730087

Cited by 28 publications

(26 citation statements)

References 7 publications

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“…The function s → α ik s − τ β ik στ p(s) is non decreasing thanks to (24) and the hypotheses on the function p. By the induction hypothesis on i, we have that…”

Section: Runge-kutta Schemesmentioning

confidence: 99%

“…Linear approximation schemes based on the so-called nonlinear Chernoff's formula with a suitable relaxation parameter have been studied for example in [6,29,30,26] where also some energy error estimates have been investigated. Other linear approximation schemes have been introduced by Jäger, Kačur and Handlovičová [17,24]. More recently, different approaches based on kinetic schemes for degenerate parabolic systems have been considered and analyzed by Aregba-Driollet, Natalini and Tang in [2].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems

Cavalli

Naldi

Puppo

et al. 2007

SIAM J. Numer. Anal.

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Several relaxation approximations to partial differential equations have been recently proposed. Examples include conservation laws, HamiltonJacobi equations, convection-diffusion problems, gas dynamics problems. The present paper focuses onto diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations. These schemes are based on a suitable semilinear hyperbolic system with relaxation terms. High order methods are obtained by coupling ENO and WENO schemes for space discretization with IMEX schemes for time integration. Error estimates and convergence analysis are developed for semidiscrete schemes with numerical analysis for fully discrete relaxed schemes. Various numerical results in one and two dimensions illustrate the high accuracy and good properties of the proposed numerical schemes, also in the degenerate case. These schemes can be easily implemented on parallel computers and applied to more general systems of nonlinear parabolic equations in twoand three-dimensional cases.

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“…The function s → α ik s − τ β ik στ p(s) is non decreasing thanks to (24) and the hypotheses on the function p. By the induction hypothesis on i, we have that…”

Section: Runge-kutta Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems

Cavalli

Naldi

Puppo

et al. 2007

SIAM J. Numer. Anal.

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show abstract

“…This scheme was adapted to more general porous medium equations in [10,13]. More recently Jäger and Kačur [11] and Kačur [12] studied the numerical approximation of (EP ).…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of nonlinear elliptic-parabolic equations

Maitre

2002

ESAIM: M2AN

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Abstract. This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996;Halpern, 1967;Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995;Kačur, 1999).Mathematics Subject Classification. 65M12, 35K65, 35K55, 65N22.

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“…Kačur, Handlovičová and Kačurová in [4] replaced the parameter µ in the linear approximation scheme by a function µ(x). This modification allows to increase the step of discretization of the time axis.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper the linear approximation scheme, introduced in [4], is used to solve the two-phase Stefan problem on a domain consisting of two components with im-perfect contact. A modification of the iterative method to determine the function µ(x) is also presented.…”

Section: Introductionmentioning

confidence: 99%

The use of linear approximation scheme for solving the Stefan problem

Dzurenda

1997

Applications of Mathematics

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Solution of Nonlinear Diffusion Problems by Linear Approximation Schemes

Cited by 28 publications

References 7 publications

High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems

High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems

Numerical analysis of nonlinear elliptic-parabolic equations

The use of linear approximation scheme for solving the Stefan problem

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