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February 2019, 12(1): 59-77. doi: 10.3934/krm.2019003

## A stochastic algorithm without time discretization error for the Wigner equation

 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

Received  June 2017 Published  July 2018

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.

Citation: Orazio Muscato, Wolfgang Wagner. A stochastic algorithm without time discretization error for the Wigner equation. Kinetic & Related Models, 2019, 12 (1) : 59-77. doi: 10.3934/krm.2019003
##### 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.

show all references

##### 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.
Density (37) (calculated with parameters (44) in which $N_k = 400$) and reference solution (39)
Density (37) (calculated with parameters (44) and $N_k = 100$) and reference solution (39)
Wave-vector density (38) (calculated with parameters (44) and $N_k = 100$) and stochastic reference solution ($N_k = 400$)
Total average position (32)
Total average velocity (33)
Total average kinetic energy (34)
Density (37) (calculated with parameters (44) and $N_x = 100$) and reference solution (39)
Numbers of particles before and after cancellations for the parameters (44)
Density (37) (calculated with parameters (44) and various $\Delta t$) and reference solution (39)
Function (47)
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
 $N_k$ $N_x$ $N_{\rm ini}$ $N_{\rm canc}$ 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
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
 $\Delta t$ (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
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