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Free Boundary Problem (free + boundary_problem)
Selected AbstractsBoundary Perturbation Methods for Water WavesGAMM - MITTEILUNGEN, Issue 1 2007David P. Nicholls Abstract The most successful equations for the modeling of ocean wave phenomena are the free,surface Euler equations. Their solutions accurately approximate a wide range of physical problems from open,ocean transport of pollutants, to the forces exerted upon oil platforms by rogue waves, to shoaling and breaking of waves in nearshore regions. These equations provide numerous challenges for theoreticians and practitioners alike as they couple the difficulties of a free boundary problem with the subtle balancing of nonlinearity and dispersion in the absence of dissipation. In this paper we give an overview of, what we term, "Boundary Perturbation" methods for the analysis and numerical simulation of this "water wave problem". Due to our own research interests this review is focused upon the numerical simulation of traveling water waves, however, the extensive literature on the initial value problem and additional theoretical developments are also briefly discussed. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] A dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2005A. P. Peirce Abstract We present a novel multigrid (MG) procedure for the efficient solution of the large non-symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi-layered elastic medium. The governing equations involve a highly non-linear coupled system of integro-partial differential equations along with the fracture front free boundary problem. The conditioning of the algebraic equations typically degrades as O(N3). A number of characteristics of this problem present significant new challenges for designing an effective MG strategy. Large changes in the coefficients of the PDE are dealt with by taking the appropriate harmonic averages of the discrete coefficients. Coarse level Green's functions for multiple elastic layers are constructed using a single dual mesh and superposition. Coarse grids that are sub-sets of the finest grid are used to treat mixed variable problems associated with ,pinch points.' Localized approximations to the Jacobian at each MG level are used to devise efficient Gauss,Seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements. The performance of the MG preconditioner is demonstrated in a number of numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd. [source] Evolution free boundary problem for equations of viscous compressible heat-conducting capillary fluidsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2001Ewa Zadrzy In the paper the global motion of a viscous compressible heat conducting capillary fluid in a domain bounded by a free surface is considered. Assuming that the initial data are sufficiently close to a constant state and the external force vanishes we prove the existence of a global-in-time solution which is close to the constant state for any moment of time. The solution is obtained in such Sobolev,Slobodetskii spaces that the velocity, the temperature and the density of the fluid have $W_2^{2+\alpha,1+\alpha/2}$\nopagenumbers\end, $W_2^{2+\alpha,1+\alpha/2}$\nopagenumbers\end and $W_2^{1+\alpha,1/2+\alpha/2}$\nopagenumbers\end ,regularity with ,,(¾, 1), respectively. Copyright © 2001 John Wiley & Sons, Ltd. [source] Asymptotic behaviour for a non-monotone fluid in one dimension: the positive temperature caseMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2001B. Ducomet We consider a one-dimensional continuous model of neutron star, described by a compressible Navier,Stokes system with a non-monotone equation of state, due to the effective Skyrme nuclear interaction between particles. We study the asymptotic behaviour of globally defined solutions of a mixed free boundary problem for our model, for large time, assuming that a sufficient thermal dissipation is present. Copyright © 2001 John Wiley & Sons, Ltd. [source] Existence and uniqueness of weak solutions for precipitation fronts: A novel hyperbolic free boundary problem in several space variablesCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2010Andrew J. Majda The determination of the large-scale boundaries between moist and dry regions is an important problem in contemporary meteorology. These phenomena have been addressed recently in a simplified tropical climate model through a novel hyperbolic free boundary formulation yielding three families (drying, slow moistening, and fast moistening) of precipitation fronts. The last two wave types violate Lax's shock inequalities yet are robustly realized. This formal hyperbolic free boundary problem is given here a rigorous mathematical basis by establishing the existence and uniqueness of suitable weak solutions arising in the zero relaxation limit. A new L2 -contraction estimate is also established at positive relaxation values. © 2010 Wiley Periodicals, Inc. [source] Local existence for the free boundary problem for nonrelativistic and Relativistic compressible Euler equations with a vacuum boundary conditionCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2009Yuri Trakhinin We study the free boundary problem for the equations of compressible Euler equations with a vacuum boundary condition. Our main goal is to recover in Eulerian coordinates the earlier well-posedness result obtained by Lindblad [11] for the isentropic Euler equations and extend it to the case of full gas dynamics. For technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain. We prove the local-in-time existence in Sobolev spaces by the technique applied earlier to weakly stable shock waves and characteristic discontinuities [5, 12]. It contains, in particular, the reduction to a fixed domain, using the "good unknown" of Alinhac [1], and a suitable Nash-Moser-type iteration scheme. A certain modification of such an approach is caused by the fact that the symbol associated to the free surface is not elliptic. This approach is still directly applicable to the relativistic version of our problem in the setting of special relativity, and we briefly discuss its extension to general relativity. © 2009 Wiley Periodicals, Inc. [source] Geometry and a priori estimates for free boundary problems of the Euler's equationCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2008Jalal Shatah In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to 0, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension. © 2007 Wiley Periodicals, Inc. [source] A free boundary problem with optimal transportationCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 1 2004Ovidiu Savin First page of article [source] Coupling finite difference methods and integral formulas for elliptic problems arising in fluid mechanicsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2004C. Albuquerque Abstract This article is devoted to the numerical analysis of two classes of iterative methods that combine integral formulas with finite-difference Poisson solvers for the solution of elliptic problems. The first method is in the spirit of the Schwarz domain decomposition method for exterior domains. The second one is motivated by potential calculations in free boundary problems and can be viewed as a numerical analytic continuation algorithm. Numerical tests are presented that confirm the convergence properties predicted by numerical analysis. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 199,229, 2004 [source] Geometry and a priori estimates for free boundary problems of the Euler's equationCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2008Jalal Shatah In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to 0, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension. © 2007 Wiley Periodicals, Inc. [source] | ||||||||||