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A comparative study on nonlocal diffusion operators related to the fractional Laplacian
An FEM-MLMC algorithm for a moving shutter diffraction in time stochastic model
1. | Department of Applied Mathematics & Statistics, Colorado School of Mines, Golden, CO 80401, USA |
2. | National Renewable Energy Laboratory, Golden, CO 80401, USA |
We consider a moving shutter and non-deterministic generalization of the diffraction in time model introduced by Moshinsky several decades ago to study a class of quantum transients. We first develop a moving-mesh finite element method (FEM) to simulate the determisitic version of the model. We then apply the FEM and multilevel Monte Carlo (MLMC) algorithm to the stochastic moving-domain model for simulation of approximate statistical moments of the density profile of the stochastic transients.
References:
[1] |
C. J. Budd, W. Huang and R. D. Russell,
Adaptivity with moving grids, Acta Numerica, 18 (2009), 111-241.
|
[2] |
A. del Campo, G. Garcia-Calderón and J. Muga,
Quantum transients, Phys. Reports, 476 (2009), 1-50.
|
[3] |
J. Dick, F. Y. Kuo, Q. L. Gia and C. Schwab,
Multilevel higher order QMC Petrov-Galerkin discretization for affine parametric operator equations, SIAM J. NUMER. ANAL., 54 (2016), 2541-2568.
doi: 10.1137/16M1078690. |
[4] |
M. Giles,
Multilevel Monte Carlo methods, Acta Numerica, 24 (2015), 259-328.
doi: 10.1017/S096249291500001X. |
[5] |
P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004. |
[6] |
A. Goussev, Diffraction in time: An exactly solvable model, Phys. Review A, 87 (2012), 053621.
doi: 10.1103/PhysRevA.87.053621. |
[7] |
T. Kimura, N. Sato and S. Iwata,
Application of the higher order finite-element method to one-dimensional Schrödinger equation, J. Comput. Chem., 9 (1998), 827-835.
doi: 10.1002/jcc.540090805. |
[8] |
T. E. Lee, M. J. Baines and S. Langdon,
A finite difference moving mesh method based on conservation for moving boundary problems, J. Comput. Appl. Math., 288 (2015), 1-17.
doi: 10.1016/j.cam.2015.03.032. |
[9] |
O. P. L. Maître and O. M. Kino, Spectral Methods for Uncertainty Quantification, Springer, 2010. |
[10] |
M. Moshinsky,
Diffraction in time, Phys. Review, 88 (1952), 625-631.
doi: 10.1103/PhysRev.88.625. |
[11] |
A. Nissen, G. Kreiss and M. Gerritsen,
High order stable finite difference methods for the Schrödinger equation, J. Sci. Computing, 55 (2013), 173-199.
doi: 10.1007/s10915-012-9628-1. |
[12] |
Z. Romanowski,
Application of h-adaptive, high order finite element method to solve radial Schrödinger equation, Molecular Physics, 107 (2009), 1339-1348.
doi: 10.1080/00268970902873554. |
show all references
References:
[1] |
C. J. Budd, W. Huang and R. D. Russell,
Adaptivity with moving grids, Acta Numerica, 18 (2009), 111-241.
|
[2] |
A. del Campo, G. Garcia-Calderón and J. Muga,
Quantum transients, Phys. Reports, 476 (2009), 1-50.
|
[3] |
J. Dick, F. Y. Kuo, Q. L. Gia and C. Schwab,
Multilevel higher order QMC Petrov-Galerkin discretization for affine parametric operator equations, SIAM J. NUMER. ANAL., 54 (2016), 2541-2568.
doi: 10.1137/16M1078690. |
[4] |
M. Giles,
Multilevel Monte Carlo methods, Acta Numerica, 24 (2015), 259-328.
doi: 10.1017/S096249291500001X. |
[5] |
P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004. |
[6] |
A. Goussev, Diffraction in time: An exactly solvable model, Phys. Review A, 87 (2012), 053621.
doi: 10.1103/PhysRevA.87.053621. |
[7] |
T. Kimura, N. Sato and S. Iwata,
Application of the higher order finite-element method to one-dimensional Schrödinger equation, J. Comput. Chem., 9 (1998), 827-835.
doi: 10.1002/jcc.540090805. |
[8] |
T. E. Lee, M. J. Baines and S. Langdon,
A finite difference moving mesh method based on conservation for moving boundary problems, J. Comput. Appl. Math., 288 (2015), 1-17.
doi: 10.1016/j.cam.2015.03.032. |
[9] |
O. P. L. Maître and O. M. Kino, Spectral Methods for Uncertainty Quantification, Springer, 2010. |
[10] |
M. Moshinsky,
Diffraction in time, Phys. Review, 88 (1952), 625-631.
doi: 10.1103/PhysRev.88.625. |
[11] |
A. Nissen, G. Kreiss and M. Gerritsen,
High order stable finite difference methods for the Schrödinger equation, J. Sci. Computing, 55 (2013), 173-199.
doi: 10.1007/s10915-012-9628-1. |
[12] |
Z. Romanowski,
Application of h-adaptive, high order finite element method to solve radial Schrödinger equation, Molecular Physics, 107 (2009), 1339-1348.
doi: 10.1080/00268970902873554. |




EOC of |
||
80 | 3.1601e-01 | |
160 | 1.3607e-01 | 1.2156 |
320 | 5.3584e-02 | 1.3445 |
640 | 2.1350e-02 | 1.3276 |
1280 | 1.0176e-02 | 1.0691 |
2560 | 4.9566e-03 | 1.0377 |
5120 | 1.7975e-03 | 1.4633 |
10240 | 3.8505e-04 | 2.2229 |
EOC of |
||
80 | 3.1601e-01 | |
160 | 1.3607e-01 | 1.2156 |
320 | 5.3584e-02 | 1.3445 |
640 | 2.1350e-02 | 1.3276 |
1280 | 1.0176e-02 | 1.0691 |
2560 | 4.9566e-03 | 1.0377 |
5120 | 1.7975e-03 | 1.4633 |
10240 | 3.8505e-04 | 2.2229 |
4.6156 | 4.6199 | 4.6282 | 4.6289 |
4.6156 | 4.6199 | 4.6282 | 4.6289 |
4.6256 | 4.6307 | 4.6276 | 4.6276 |
4.6256 | 4.6307 | 4.6276 | 4.6276 |
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