| 1. |
- Engquist, Björn, et al.
(författare)
-
Projection shock capturing algorithms
- 1990
-
Ingår i: Twelfth International Conference on Numerical Methods in Fluid Dynamics. - Berlin : Springer-Verlag. ; , s. 335-336
-
Konferensbidrag (refereegranskat)
|
|
| 2. |
- Häggblad, Jon, 1981-
(författare)
-
Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method
- 2012
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis centers on modified stencils for the Finite-Difference Time-Domain method (FDTD), or Yee scheme, when modelling curved boundaries, obstacles and holes smaller than the discretization length. The goal is to increase the accuracy while keeping the structure of the standard method, enabling improvements to existing implementations with minimal effort.We present an extension of a previously developed technique for consistent boundary approximation in the Yee scheme. We consider both Maxwell's equations and the acoustic equations in three dimensions, which require separate treatment, unlike in two dimensions.The stability properties of coefficient modifications are essential for practical usability. We present an analysis of the requirements for time-stable modifications, which we use to construct a simple and effective method for boundary approximations. The method starts from a predetermined staircase discretization of the boundary, requiring no further data on the underlying geometry that is being approximated.Not only is the standard staircasing of curved boundaries a poor approximation, it is inconsistent, giving rise to errors that do not disappear in the limit of small grid lengths. We analyze the standard staircase approximation by deriving exact solutions of the difference equations, including the staircase boundary. This facilitates a detailed error analysis, showing how staircasing affects amplitude, phase, frequency and attenuation of waves.To model obstacles and holes of smaller size than the grid length, we develop a numerical subgrid method based on locally modified stencils, where a highly resolved micro problem is used to generate effective coefficients for the Yee scheme at the macro scale.The implementations and analysis of the developed methods are validated through systematic numerical tests.
|
|
| 3. |
- Abdulle, A., et al.
(författare)
-
Finite element heterogeneous multiscale methods with near optimal computational complexity
- 2007
-
Ingår i: Multiscale Modeling & simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3459 .- 1540-3467. ; 6:4, s. 1059-1084
-
Tidskriftsartikel (refereegranskat)abstract
- This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the heterogeneous multiscale methods (HMM), we propose a micro-macro approach which combines the finite element method (FEM) for the macroscopic solver and the pseudospectral method for the microsolver. Unlike the micro-macromethods based on the standard FEM proposed so far, in the HMM we obtain, for periodic homogenization problems, a method that (slow) variable.
|
|
| 4. |
- Andersson, U, et al.
(författare)
-
A contribution to wavelet-based subgrid modeling
- 1999
-
Ingår i: Applied and Computational Harmonic Analysis. - 1063-5203 .- 1096-603X. ; 7:2, s. 151-164
-
Tidskriftsartikel (refereegranskat)abstract
- A systematic technique for the derivation of subgrid scale models in the numerical solution of partial differential equations is described. The technique is based on Haar wavelet projections of the discrete operator followed by a sparse approximation. As numerical testing suggests, the resulting numerical method will accurately represent subgrid scale phenomena on a coarse grid. Applications to numerical homogenization and wave propagation in materials with subgrid inhomogeneities are presented.
|
|
| 5. |
- Ariel, Gil, et al.
(författare)
-
A multiscale method for stiff ordinary differential equations with resonance
- 2009
-
Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 78:266, s. 929-956
-
Tidskriftsartikel (refereegranskat)abstract
- A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam.
|
|
| 6. |
- Ariel, Gil, et al.
(författare)
-
A reversible multiscale integration method
- 2009
-
Ingår i: Communications in Mathematical Sciences. - : Duke University Press. - 1539-6746 .- 1945-0796. ; 7:3, s. 595-610
-
Tidskriftsartikel (refereegranskat)abstract
- A multiscale, time reversible method for computing the effective slow behavior of systems of highly oscillatory ordinary differential equations is presented. The proposed method relies on correctly tracking a set of slow variables that is sufficient to approximate any variable and functional that are slow under the dynamics of the system. The algorithm follows the framework of the heterogeneous multiscale method. The notion of time reversibility in the multiple time-scale setting is discussed. The algorithm requires nontrivial matching between the microscopic state variables and the macroscopic slow ones. Numerical examples show the efficiency of the multiscale method and the advantages of time reversibility.
|
|
| 7. |
- Ariel, Gil, et al.
(författare)
-
Gaussian Beam Decomposition of High Frequency Wave Fields Using Expectation-Maximization
- 2011
-
Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 230:6, s. 2303-2321
-
Tidskriftsartikel (refereegranskat)abstract
- A new numerical method for approximating highly oscillatory wave fields as a superposition of Gaussian beams is presented. The method estimates the number of beams and their parameters automatically. This is achieved by an expectation–maximization algorithm that fits real, positive Gaussians to the energy of the highly oscillatory wave fields and its Fourier transform. Beam parameters are further refined by an optimization procedure that minimizes the difference between the Gaussian beam superposition and the highly oscillatory wave field in the energy norm.
|
|
| 8. |
- Ariel, G., et al.
(författare)
-
Multiscale computations for highly oscillatory problems
- 2009
-
Ingår i: Multiscale Modeling and Simulation in Science. - Berlin, Heidelberg : Springer Berlin/Heidelberg. ; , s. 237-287
-
Konferensbidrag (refereegranskat)abstract
- We review a selection of essential techniques for constructing computational multiscale methods for highly oscillatory ODEs. Contrary to the typical approaches that attempt to enlarge the stability region for specialized problems, these lecture notes emphasize how multiscale properties of highly oscillatory systems can be characterized and approximated in a truly multiscale fashion similar to the settings of averaging and homogenization. Essential concepts such as resonance, fast-slow scale interactions, averaging, and techniques for transformations to non-stiff forms are discussed in an elementary manner so that the materials can be easily accessible to beginning graduate students in applied mathematics or computational sciences.
|
|
| 9. |
|
|
| 10. |
- Ariel, Gil, et al.
(författare)
-
Numerical multiscale methods for coupled oscillators
- 2009
-
Ingår i: Multiscale Modeling & simulation. - : Society for Industrial & Applied Mathematics (SIAM). - 1540-3459 .- 1540-3467. ; 7:3, s. 1387-1404
-
Tidskriftsartikel (refereegranskat)abstract
- A multiscale method for computing the effective slow behavior of a system of weakly coupled nonlinear planar oscillators is presented. The oscillators may be either in the form of a periodic solution or a stable limit cycle. Furthermore, the oscillators may be in resonance with one another and thereby generate some hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables that is sufficient to approximate any variable and functional that are slow under the dynamics of the ordinary differential equation. The technique is more efficient than existing methods, and its advantages are demonstrated with examples. The algorithm follows the framework of the heterogeneous multiscale method.
|
|
