21,377 research outputs found
Moving Mesh Methods for Problems with Blow-Up
This is the published version, also available here: http://dx.doi.org/10.1137/S1064827594272025.In this paper we consider the numerical solution of PDEs with blow-up for which scaling invariance plays a natural role in describing the underlying solution structures. It is a challenging numerical problem to capture the qualitative behaviour in the blow-up region, and the use of nonuniform meshes is essential. We consider moving mesh methods for which the mesh is determined using so-called moving mesh partial differential equations (MMPDEs).Specifically, the underlying PDE and the MMPDE are solved for the blow-up solution and the computational mesh simultaneously. Motivated by the desire for the MMPDE to preserve the scaling invariance of the underlying problem, we study the effect of different choices of MMPDEs and monitor functions. It is shown that for suitable ones the MMPDE solution evolves towards a. (moving) mesh which close to the blow-up point automatically places the mesh points in such a manner that the ignition kernel, which is well known to be a natural coordinate in describing the behaviour of blow-up, approaches a constant as (the blow-up time). Several numerical examples are given to verify the theory for these MMPDE methods and to illustrate their efficacy
A moving mesh method with variable relaxation time
We propose a moving mesh adaptive approach for solving time-dependent partial
differential equations. The motion of spatial grid points is governed by a
moving mesh PDE (MMPDE) in which a mesh relaxation time \tau is employed as a
regularization parameter. Previously reported results on MMPDEs have invariably
employed a constant value of the parameter \tau. We extend this standard
approach by incorporating a variable relaxation time that is calculated
adaptively alongside the solution in order to regularize the mesh appropriately
throughout a computation. We focus on singular problems involving self-similar
blow-up to demonstrate the advantages of using a variable relaxation ime over a
fixed one in terms of accuracy, stability and efficiency.Comment: 21 page
Numerical investigations of traveling singular sources problems via moving mesh method
This paper studies the numerical solution of traveling singular sources
problems. In such problems, a big challenge is the sources move with different
speeds, which are described by some ordinary differential equations. A
predictor-corrector algorithm is presented to simulate the position of singular
sources. Then a moving mesh method in conjunction with domain decomposition is
derived for the underlying PDE. According to the positions of the sources, the
whole domain is splitted into several subdomains, where moving mesh equations
are solved respectively. On the resulting mesh, the computation of jump
is avoided and the discretization of the underlying PDE is reduced
into only two cases. In addition, the new method has a desired second-order of
the spatial convergence. Numerical examples are presented to illustrate the
convergence rates and the efficiency of the method. Blow-up phenomenon is also
investigated for various motions of the sources
Structures and waves in a nonlinear heat-conducting medium
The paper is an overview of the main contributions of a Bulgarian team of
researchers to the problem of finding the possible structures and waves in the
open nonlinear heat conducting medium, described by a reaction-diffusion
equation. Being posed and actively worked out by the Russian school of A. A.
Samarskii and S.P. Kurdyumov since the seventies of the last century, this
problem still contains open and challenging questions.Comment: 23 pages, 13 figures, the final publication will appear in Springer
Proceedings in Mathematics and Statistics, Numerical Methods for PDEs:
Theory, Algorithms and their Application
A moving mesh method for one-dimensional hyperbolic conservation laws
We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work
Symmetry-preserving discrete schemes for some heat transfer equations
Lie group analysis of differential equations is a generally recognized
method, which provides invariant solutions, integrability, conservation laws
etc. In this paper we present three characteristic examples of the construction
of invariant difference equations and meshes, where the original continuous
symmetries are preserved in discrete models. Conservation of symmetries in
difference modeling helps to retain qualitative properties of the differential
equations in their difference counterparts.Comment: 21 pages, 4 ps figure
Non-self-similar blow-up in the heat flow for harmonic maps in higher dimensions
We analyze the finite-time blow-up of solutions of the heat flow for
-corotational maps . For each dimension
we construct a countable family of blow-up solutions via a
method of matched asymptotics by glueing a re-scaled harmonic map to the
singular self-similar solution: the equatorial map. We find that the blow-up
rates of the constructed solutions are closely related to the eigenvalues of
the self-similar solution. In the case of -corotational maps our solutions
are stable and represent the generic blow-up.Comment: 26 pages, 5 figure
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