391 research outputs found
A space-time discretization of a nonlinear peridynamic model on a 2D lamina
Peridynamics is a nonlocal theory for dynamic fracture analysis consisting in
a second order in time partial integro-differential equation. In this paper, we
consider a nonlinear model of peridynamics in a two-dimensional spatial domain.
We implement a spectral method for the space discretization based on the
Fourier expansion of the solution while we consider the Newmark- method
for the time marching. This computational approach takes advantages from the
convolutional form of the peridynamic operator and from the use of the discrete
Fourier transform. We show a convergence result for the fully discrete
approximation and study the stability of the method applied to the linear
peridynamic model. Finally, we perform several numerical tests and comparisons
to validate our results and provide simulations implementing a volume
penalization technique to avoid the limitation of periodic boundary conditions
due to the spectral approach
An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems
In this paper we present an asymptotically compatible meshfree method for
solving nonlocal equations with random coefficients, describing diffusion in
heterogeneous media. In particular, the random diffusivity coefficient is
described by a finite-dimensional random variable or a truncated combination of
random variables with the Karhunen-Lo\`{e}ve decomposition, then a
probabilistic collocation method (PCM) with sparse grids is employed to sample
the stochastic process. On each sample, the deterministic nonlocal diffusion
problem is discretized with an optimization-based meshfree quadrature rule. We
present rigorous analysis for the proposed scheme and demonstrate convergence
for a number of benchmark problems, showing that it sustains the asymptotic
compatibility spatially and achieves an algebraic or sub-exponential
convergence rate in the random coefficients space as the number of collocation
points grows. Finally, to validate the applicability of this approach we
consider a randomly heterogeneous nonlocal problem with a given spatial
correlation structure, demonstrating that the proposed PCM approach achieves
substantial speed-up compared to conventional Monte Carlo simulations
A finite difference method for fractional diffusion equations with Neumann boundary conditions
A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension.
The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The well-posedness of the obtained initial value problem is proved and it is pointed out that each extensions is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted GrĂĽnwald--Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved
Blowup in diffusion equations: A survey
AbstractThis paper deals with quasilinear reaction-diffusion equations for which a solution local in time exists. If the solution ceases to exist for some finite time, we say that it blows up. In contrast to linear equations blowup can occur even if the data are smooth and well-defined for all times. Depending on the equation either the solution or some of its derivatives become singular. We shall concentrate on those cases where the solution becomes unbounded in finite time. This can occur in quasilinear equations if the heat source is strong enough. There exist many theoretical studies on the question on the occurrence of blowup. In this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power. A better knowledge would be useful also for their numerical treatment. Thus, not surprisingly, the numerical analysis of this type of problems is still at a rather early stage. The goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems
Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations
In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations
- …