Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation
SIAM Journal on Applied Mathematics
Homogenization and two-scale convergence
SIAM Journal on Mathematical Analysis
A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Wavelet-Based Numerical Homogenization
SIAM Journal on Numerical Analysis
Hybrid atomistic-continuum formulations and the moving contact-line problem
Journal of Computational Physics
Computing
Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum
SIAM Journal on Scientific Computing
Constraint-Defined Manifolds: a Legacy Code Approach to Low-Dimensional Computation
Journal of Scientific Computing
Patch Dynamics for Multiscale Problems
Computing in Science and Engineering
A hybrid molecular continuum method using point wise coupling
Advances in Engineering Software
| Hi-index | 31.45 |
An important class of problems exhibits smooth behaviour on macroscopic space and time scales, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, an ''equation-free'' framework has been proposed, of which patch dynamics is an essential component. Patch dynamics is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes (patches), which cover only a fraction of the space-time domain. We show that it is possible to use arbitrary boundary conditions for these patches, provided that suitably large buffer regions ''shield'' the boundary artefacts from the interior of the patches. We analyze the accuracy of this scheme for a diffusion homogenization problem with periodic heterogeneity and illustrate the approach with a set of numerical examples, which include a non-linear reaction-diffusion equation and the Kuramoto-Sivashinsky equation.