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This article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations (PDEs) in several spatial dimensions. The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation. This method gives a new technique for performing r-adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a (time-varying) scalar monitor function (used successfully in moving mesh methods in one-dimension). Detailed analysis of this new method is presented in which the convergence, regularity, and stability of the mesh is studied. Additionally, this new method is shown to be straightforward to program and implement, requiring the solution of only one simple scalar time-dependent equation in arbitrary dimension, with adaptivity along the boundaries handled automatically. We discuss three preexisting methods in the context of this work. Examples are presented in which either the monitor function is prescribed in advance, or it is given by the solution of a partial differential equation. [ABSTRACT FROM AUTHOR]

In this paper we develop and analyze a moving mesh code for the adaptive simulation of fourth-order PDEs based on collocation. The scheme is shown to enforce discrete conservation for problems written in a generalized conservation form. To demonstrate the breadth of applicability we present examples from both Cahn--Hilliard and thin-film--type equations exhibiting metastable behavior, finite-time solution blow-up, finite-time extinction, and moving interfaces. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of classes of global and blow-up solutions of a semilinear parabolic equation of the "limit" Cahn-Hilliard type ut = -Δ(Δu + |u|p-1 u) in RN x R+, p > 1, with bounded integrable initial data. We show that in some {p,N}-parameter ranges it admits a countable set of blow-up similarity patterns. The most interesting set of blow-up solutions is constructed at the first critical exponent p = p0 = 1 + 2/N, where the first simplest profile is shown to be stable. Unlike the blow-up case, we show that, for p = p0, the set of global decaying source-type similarity solutions is continuous and determine the stable mass-branch. We prove that there exists a countable spectrum of critical exponents {p = pl = 1 + 2/N+l, l = 0, 1, 2, ...} creating bifurcation branches, which play a key role in general description of solutions globally decaying as t → ∞. [ABSTRACT FROM AUTHOR]

In this article we construct, both asymptotically and numerically, multibump, blow-up, self-similar solutions to the complex Ginzburg--Landau equation (CGL) in the limit of small dissipation. Through a careful asymptotic analysis, involving a balance of both algebraic and exponential terms, we determine the parameter range over which these solutions may exist. Most intriguingly, we determine a branch of solutions that are not perturbations of solutions to the nonlinear Schrödinger equation (NLS); moreover, they are not monotone, but they are stable. Furthermore, these axisymmetric ring-like solutions exist over a broader parameter regime than the monotone profile. [ABSTRACT FROM AUTHOR]