2,411 research outputs found
Global existence of the harmonic map heat flow into Lorentzian manifolds
We investigate a parabolic-elliptic system for maps from a compact
Riemann surface into a Lorentzian manifold with a
warped product metric. That system turns the harmonic map type equations into a
parabolic system, but keeps the -equation as a nonlinear second order
constraint along the flow. We prove a global existence result of the
parabolic-elliptic system by assuming either some geometric conditions on the
target Lorentzian manifold or small energy of the initial maps. The result
implies the existence of a Lorentzian harmonic map in a given homotopy class
with fixed boundary data.Comment: to appear in J. Math. Pures App
A RBF partition of unity collocation method based on finite difference for initial-boundary value problems
Meshfree radial basis function (RBF) methods are popular tools used to
numerically solve partial differential equations (PDEs). They take advantage of
being flexible with respect to geometry, easy to implement in higher
dimensions, and can also provide high order convergence. Since one of the main
disadvantages of global RBF-based methods is generally the computational cost
associated with the solution of large linear systems, in this paper we focus on
a localizing RBF partition of unity method (RBF-PUM) based on a finite
difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation
method, which can successfully be applied to solve time-dependent PDEs. This
approach allows to significantly decrease ill-conditioning of traditional
RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix
system, reducing the computational effort but maintaining at the same time a
high level of accuracy. Numerical experiments show performances of our
collocation scheme on two benchmark problems, involving unsteady
convection-diffusion and pseudo-parabolic equations
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