4 research outputs found
Pointwise a posteriori error bounds for blow-up in the semilinear heat equation
This work is concerned with the development of a space-time adaptive
numerical method, based on a rigorous a posteriori error bound, for the
semilinear heat equation with a general local Lipschitz reaction term whose
solution may blow-up in finite time. More specifically, conditional a
posteriori error bounds are derived in the norm for a
first order in time, implicit-explicit (IMEX), conforming finite element method
in space discretization of the problem. Numerical experiments applied to both
blow-up and non blow-up cases highlight the generality of our approach and
complement the theoretical results
A Structure Preserving Scheme for the Kolmogorov-Fokker-Planck Equation
In this paper we introduce a numerical scheme which preserves the long time
behavior of solutions to the Kolmogorov equation. The method presented is based
on a self-similar change of variables technique to transform the Kolmogorov
equation into a new form, such that the problem of designing structure
preserving schemes, for the original equation, amounts to building a standard
scheme for the transformed equation. We also present an analysis for the
operator splitting technique for the self-similar method and numerical results
for the described scheme
A mass-lumping finite element method for radially symmetric solution of a multidimensional semilinear heat equation with blow-up
This study presents a new mass-lumping finite element method for computing
the radially symmetric solution of a semilinear heat equation in an
dimensional ball (). We provide two schemes, (ML-1) and (ML-2), and
derive their error estimates through the discrete maximum principle. In the
weighted norm, the convergence of (ML-1) was at the optimal order but
that of (ML-2) was only at sub-optimal order. Nevertheless, scheme (ML-2)
reproduces a blow-up of the solution of the original equation. In fact, in
scheme (ML-2), we could accurately approximate the blow-up time. Our
theoretical results were validated in numerical experiments.Comment: 28 pages, 10 figure
Potentially Singular Solutions of the 3D Incompressible Euler Equations
Whether the 3D incompressible Euler equations can develop a singularity in
finite time from smooth initial data is one of the most challenging problems in
mathematical fluid dynamics. This work attempts to provide an affirmative
answer to this long-standing open question from a numerical point of view, by
presenting a class of potentially singular solutions to the Euler equations
computed in axisymmetric geometries. The solutions satisfy a periodic boundary
condition along the axial direction and no-flow boundary condition on the solid
wall. The equations are discretized in space using a hybrid 6th-order Galerkin
and 6th-order finite difference method, on specially designed adaptive (moving)
meshes that are dynamically adjusted to the evolving solutions. With a maximum
effective resolution of over near the point of the
singularity, we are able to advance the solution up to
and predict a singularity time of , while achieving a
\emph{pointwise} relative error of in the vorticity vector
and observing a -fold increase in the maximum
vorticity . The numerical data are checked against all
major blowup (non-blowup) criteria, including Beale-Kato-Majda,
Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the
singularity. A local analysis near the point of the singularity also suggests
the existence of a self-similar blowup in the meridian plane.Comment: version 1: 81 pages, 86 figures; version 2: 57 pages, 44 figures,
removed part of the technical details and streamlined the presentatio