A stochastic algorithm without time discretization error for the Wigner equation

Orazio Muscato; Wolfgang Wagner

doi:10.3934/krm.2019003

Article Contents

2019, Volume 12, Issue 1: 59-77. Doi: 10.3934/krm.2019003

This issue Previous Article Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density Next Article Numerical solutions for multidimensional fragmentation problems using finite volume methods

A stochastic algorithm without time discretization error for the Wigner equation

Orazio Muscato^1, , and
Wolfgang Wagner^2,

1.
Dipartimento di Matematica e Informatica, Università degli studi di Catania, viale A. Doria 6, 95125 Catania, Italy
2.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39 - 10117 Berlin, Germany

^* Corresponding author: Orazio Muscato
^* Corresponding author: Orazio Muscato

Received: May 31, 2017

Published: July 2018

Abstract Full Text(HTML) Figure(10) / Table(2) Related Papers Cited by

Abstract

Stochastic particle methods for the numerical treatment of the Wigner equation are considered. The approximation properties of these methods depend on several numerical parameters. Such parameters are the number of particles, a time step (if transport and other processes are treated separately) and the grid size (used for the discretization of the position and the wave-vector). A stochastic algorithm without time discretization error is introduced. Its derivation is based on the theory of piecewise deterministic Markov processes. Numerical experiments are performed in a one-dimensional test case. Approximation properties with respect to the grid size and the number of particles are studied. Convergence of a time-splitting scheme to the no-splitting algorithm is demonstrated. The no-splitting algorithm is shown to be more efficient in terms of computational effort.

Keywords:

Mathematics Subject Classification: 65C05, 81S30, 82D37.

Citation:

Full Text(HTML)

Figure 1. Density (37) (calculated with parameters (44) in which $N_k = 400$) and reference solution (39)

Download: Full-size image PowerPoint slide

Figure 2. Density (37) (calculated with parameters (44) and $N_k = 100$) and reference solution (39)

Download: Full-size image PowerPoint slide

Figure 3. Wave-vector density (38) (calculated with parameters (44) and $N_k = 100$) and stochastic reference solution ($N_k = 400$)

Download: Full-size image PowerPoint slide

Figure 4. Total average position (32)

Download: Full-size image PowerPoint slide

Figure 5. Total average velocity (33)

Download: Full-size image PowerPoint slide

Figure 6. Total average kinetic energy (34)

Download: Full-size image PowerPoint slide

Figure 7. Density (37) (calculated with parameters (44) and $N_x = 100$) and reference solution (39)

Download: Full-size image PowerPoint slide

Figure 8. Numbers of particles before and after cancellations for the parameters (44)

Download: Full-size image PowerPoint slide

Figure 9. Density (37) (calculated with parameters (44) and various $\Delta t$) and reference solution (39)

Download: Full-size image PowerPoint slide

Figure 10. Function (47)

Download: Full-size image PowerPoint slide

Table 1. Properties of the algorithm for various sets of cancellation parameters. The quantity "calls" denotes the number of calls to the cancellation procedure and $N_{\rm after}$ is the average number of particles after the last cancellation

$N_k$	$N_x$	$N_{\rm ini}$	$N_{\rm canc}$	calls	$N_{\rm after}$	CPU (sec)
400	400	160k	480k	20	358k	256
100	400	160k	480k	14	273k	218
400	100	160k	480k	15	301k	229
400	400	40k	480k	7	174k	151
400	400	160k	960k	9	402k	345

| Show Table

DownLoad: CSV

Table 2. Properties of the time-splitting algorithm and the no-splitting algorithm. The quantities "err-max" and "err-aver" denote, respectively, the maximum and the average (over the cells) of the absolute differences between the measured density and the reference solution. The last column provides the measurements of the first, second and third cancellation time

$\Delta t$ (fsec.)	CPU (sec.)	err-max	err-aver	canc. times
1	278	0.0106	0.0024	3.0000, 6.0000, 8.0000
0.4	395	0.0053	0.0009	2.4000, 4.4000, 6.4000
0.1	928	0.0019	0.0004	2.0000, 3.8000, 5.4870
0.05	1628	0.0018	0.0003	1.9500, 3.6700, 5.2730
0.025	3028	0.0017	0.0003	1.9222, 3.6222, 5.2182
no-splitting	256	0.0013	0.0002	1.8927, 3.5716, 5.1428

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	A. Arnold, I. Gamba, M. Gualdani, S. Mischler, C. Mouhot and C. Sparber, The WignerFokker-Planck equation: Stationary states and large time behavior, Math. Models Methods Appl. Sci., 22 (2012), 1250034, 31pp. doi: 10.1142/S0218202512500340.
[2]	P. Ellinghaus, J. Weinbub, M. Nedjalkov, S. Selberherr and I. Dimov, Distributed-memory parallelization of the Wigner Monte Carlo method using spatial domain decomposition, J. Comput. Electronics, 14 (2015), 151-162. doi: 10.1007/s10825-014-0635-3.
[3]	I. Gamba, M. Gualdani and R. Sharp, An adaptable discontinuous Galerkin scheme for the Wigner-Fokker-Planck equation, Comm. Math. Sci., 7 (2009), 635-664. doi: 10.4310/CMS.2009.v7.n3.a7.
[4]	I. Gamba, S. Rjasanow and W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Math. Comput. Modelling, 42 (2005), 683-700. doi: 10.1016/j.mcm.2004.02.047.
[5]	C. Jacoboni and P. Bordone, The Wigner-function approach to non-equilibrium electron transport, Rep. Prog. Phys., 67 (2004), 1033-1071. doi: 10.1088/0034-4885/67/7/R01.
[6]	C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device simulation, Springer, 1989. doi: 10.1007/978-3-7091-6963-6.
[7]	O. Muscato, A benchmark study of the Signed-particle Monte Carlo algorithm for the Wigner equation, Comm. Applied Industr. Mathematics, 8 (2017), 237-250. doi: 10.1515/caim-2017-0012.
[8]	O. Muscato, V. Di Stefano and W. Wagner, A variance-reduced electrothermal Monte Carlo method for semiconductor device simulation, Comput. Math. Appl., 65 (2013), 520-527. doi: 10.1016/j.camwa.2012.03.100.
[9]	O. Muscato and W. Wagner, Time step truncation in direct simulation Monte Carlo for semiconductors, COMPEL, 24 (2005), 1351-1366. doi: 10.1108/03321640510615652.
[10]	O. Muscato and W. Wagner, A class of stochastic algorithms for the Wigner equation, SIAM J. Sci. Comput., 38 (2016), A1483-A1507. doi: 10.1137/16M105798X.
[11]	O. Muscato, W. Wagner and V. Di Stefano, Numerical study of the systematic error in Monte Carlo schemes for semiconductors, ESAIM: M2AN, 44 (2010), 1049-1068. doi: 10.1051/m2an/2010051.
[12]	M. Nedjalkov, H. Kosina, S. Selberherr, C. Ringhofer and D. Ferry, Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices, Phys. Rev. B, 70 (2004), 115319. doi: 10.1103/PhysRevB.70.115319.
[13]	M. Nedjalkov, D. Querlioz, P. Dollfus and H. Kosina, Wigner function approach, in Nano-Electronic Devices: Semiclassical and Quantum Transport Modeling, D. Vasileska and S. M. Goodnick, eds., Springer New York, 2011,289-358. doi: 10.1007/978-1-4419-8840-9_5.
[14]	D. Querlioz and P. Dollfus, The Wigner Monte Carlo Method for Nanoelectronic Devices, Wiley, 2010.
[15]	S. Rjasanow and W. Wagner, Time splitting error in DSMC schemes for the spatially homogeneous inelastic Boltzmann equation, SIAM J. Num. Anal., 45 (2007), 54-67. doi: 10.1137/050643842.
[16]	S. Shao and J. Sellier, Comparison of deterministic and stochastic methods for time-dependent Wigner simulations, J. Comput. Phys., 300 (2015), 167-185. doi: 10.1016/j.jcp.2015.08.002.
[17]	W. Wagner, A random cloud model for the Wigner equation, Kin. Related Models, 9 (2016), 217-235. doi: 10.3934/krm.2016.9.217.
[18]	E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. doi: 10.1007/978-3-642-59033-7_9.