Parabolic Monge–Ampère methods for blow-up problems in several spatial dimensions

Budd, C. J.; Williams, J. F.

doi:10.1088/0305-4470/39/19/s06

Cited by 38 publications

(57 citation statements)

References 16 publications

(22 reference statements)

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“…For α = 1 the problem is called singular with a singularity of the first kind; for α > 1 it is essentially singular (singularity of the second kind). The search for efficient numerical methods to solve (1a) is strongly motivated by numerous applications from physics [9,10,23,42], chemistry [13,39,41], mechanics [12], ecology [30,37], or economics [14,15,24]. Also, research activities in related fields, such as the computation of connecting orbits in dynamical systems [38], or singular Sturm-Liouville problems [6], benefit from techniques developed for problems of the form (1a).…”

Section: Introductionmentioning

confidence: 99%

Collocation methods for index 1 DAEs with a singularity of the first kind

et al. 2010

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Abstract. We study the convergence behavior of collocation schemes applied to approximate solutions of BVPs in linear index 1 DAEs which exhibit a critical point at the left boundary. Such a critical point of the DAE causes a singularity within the inherent ODE system. We focus our attention on the case when the inherent ODE system is singular with a singularity of the first kind, apply polynomial collocation to the original DAE system and consider different choices of the collocation points such as equidistant, Gaussian or Radau points. We show that for a well-posed boundary value problem for DAEs having a sufficiently smooth solution, the global error of the collocation scheme converges with the order O(h s ), where s is the number of collocation points. Superconvergence cannot be expected in general due to the singularity, not even for the differential components of the solution. The theoretical results are illustrated by numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Collocation methods for index 1 DAEs with a singularity of the first kind

et al. 2010

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“…[1][2][3] L2 Monge-Kantorovich theory was shown to be very promising for grid generation/adaptation applications. In this paper, we have extended the L2 Monge-Kantorovich optimization theory to Lp.…”

Section: Discussionmentioning

confidence: 99%

“…In this paper we investigate the problem of minimizing the Lp norm rJ ] l i p ~p(x) lx' -xlPdx (3) with the Jacobian constraint of Eq. (2) .…”

Section: Preprint Submitted To Elsevier Sciencementioning

confidence: 99%

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Generalized Monge–Kantorovich Optimization for Grid Generation and Adaptation in $L_{p}$

Delzanno¹,

Finn²

2010

SIAM J. Sci. Comput.

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The Monge-Kantorovich grid generation and adaptation scheme of [Delzanno et al, J. Gomput. Phys. 227, 9841 (2008)] is generalized from a variational principle based on L2 to a variational principle based on Lp. A generalized Monge-Ampere (MA) equation is derived and its properties are discussed . Results for p > 1 are obtained and compared in terms of the quality of the resulting grid. We conclude that for the grid generation application, the formulation based on Lp for p close to unity leads to serious problems associated with the boundary. Results for 1.5 :s p :s 2.5 are quite good, but there is a fairly narrow range around p = 2 where the results are close to optimal with respect to grid distortion. Furthermore, the Newton-Krylov methods used to solve the generalized MA equation perform best for p = 2.

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“…(An approximate form of the MA equation was used for grid generation in Ref. [5].) There are two sources of nonlinearity: the Hessian and the dependence of the right side on x = x+∇Φ.…”

Section: Variational Principle With Local Constraintmentioning

confidence: 99%

Grid Generation and Adaptation by Monge-Kantorovich Optimization in Two and Three Dimensions

Finn

Delzanno

Chacòn

Proceedings of the 17th International Meshing Roundtable

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Summary. The derivation of the Monge-Ampère (MA) equation, as it results from a variational principle involving grid displacement, is outlined in two dimensions (2D). This equation, a major element of Monge-Kantorovich (MK) optimization, is discussed both in the context of grid generation and grid adaptation. It is shown that grids which are generated by the MA equation also satisfy equations of an alternate variational principle minimizing grid distortion. Numerical results are shown, indicating robustness to grid tangling. Comparison is made with the deformation method [G. Liao and D. Anderson, Appl. Analysis 44, 285 (1992)], the existing method of equidistribution. A formulation is given for more general physical domains, including those with curved boundary segments. The Monge-Ampère equation is also derived in three dimensions (3D). Several numerical examples, both with more general 2D domains and in 3D, are given.

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Parabolic Monge–Ampère methods for blow-up problems in several spatial dimensions

Cited by 38 publications

References 16 publications

Collocation methods for index 1 DAEs with a singularity of the first kind

Collocation methods for index 1 DAEs with a singularity of the first kind

Generalized Monge–Kantorovich Optimization for Grid Generation and Adaptation in $L_{p}$

Grid Generation and Adaptation by Monge-Kantorovich Optimization in Two and Three Dimensions

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