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Mathematical and Computer Modelling of Dynamical Systems
Methods, Tools and Applications in Engineering and Related Sciences
Volume 20, 2014 - Issue 4
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Original Articles

Model order reduction for numerical simulation of particle transport based on numerical integration approaches

Pages 317-344 | Received 29 Mar 2012, Accepted 22 Oct 2013, Published online: 05 Dec 2013

References

  • M.A. Lieberman and A.J. Lichtenberg, Principle of Plasma Discharges and Materials Processing, 2nd ed., Wiley-Interscience, Hoboken, NJ, 2005.
  • M. Ohring, Materials Science of Thin Films, 2nd ed., Academic Press, San Diego, CA, 2002.
  • J. Geiser, V. Buck, and M. Arab, Model of PE-CVD apparatus: Verification and simulations, Math. Probl. Eng. (2010), Article ID 407561.
  • T.K. Senega and R.P. Brinkmann, A multi-component transport model for non-equilibrium low-temperature low-pressure plasmas, J. Phys. D Appl. Phys. 39 (2006), pp. 1606–1618.
  • J. Geiser and R. Roehle, Simulation of a multi-component transport model for the chemical reaction of a CVD-process, WASET, EEEng 2 (2) (2008), pp. 86–94.
  • H. Neunzert and J. Struckmeier, Particle methods for Boltzmann equation, Acta Numer. (1995), pp. 417–457.
  • C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag, Berlin, 1987.
  • M.K. Gobbert and C.A. Ringhofer, An asymptotic analysis for a model of chemical vapor deposition on a microstructured surface, SIAM J. Appl. Math. 58 (1998), pp. 737–752.
  • C. Ringhofer, Dissipative discretization methods for approximations to the Boltzmann equation, Math. Models Methods Appl. Sci. 12 (2001), pp. 133–148.
  • P.A. Raviart, An analysis of particle methods, in Numerical Methods in Fluid Dynamics, Lecture Notes in Math, Vol. 1127, F. Brezzi, ed., Springer-Verlag, Berlin, 1985, pp. 243–324.
  • R. Hockney and J. Eastwood, Computer Simulation Using Particles, Taylor and Francis, New York, 1988.
  • S. Rjasanow and W. Wagner, A generalized collision mechanism for stochastic particle schemes approximating Boltzmann type equations, Comput. Math. Appl. 35 (1998), pp. 165–178.
  • M. Asadzadech and A. Sopasakis, Convergence of a hp-streamline diffusion scheme for VlasovFokkerPlank, Math. Models. Methods. Appl. Sci. 17 (2007), pp. 1159–1182.
  • S. Rjasanow, T. Schreiber, and W. Wagner, Reduction of the number of particles in the stochastic weighted particle method for the Boltzmann equation, J. Comput. Phys 145 (1) (1998), pp. 382–405.
  • P. Crispel, P. Degond, and M.H. Vignal, A plasma expansion model based on the full Euler–Poisson model, Math. Models. Methods. Appl. Sci. 17 (2007), pp. 1129–1158.
  • S. Motta and J. Wick, A new numerical methods for kinetic equations in several dimensions, Computing 46 (1991), pp. 223–232.
  • S. Motta, A new formulation and gauge invariance of the MW-CRF method for kinetic equations, Math. Comput. Model. 36 (4–5) (2002), pp. 403–410.
  • S. Motta, Energy conservation property of the MW-CRF deterministic particle method, Appl. Math. Lett. 16 (2003), pp. 287–292.
  • C. Bianca, F. Pappalardo, and S. Motta, The MWF method for kinetic equations system, Comput. Math. Appl. 57 (2009), pp. 831–840.
  • C. Bianca and S. Motta, The MWF method: A convergence theorem for homogeneous one-dimensional case, Comput. Math. Appl. 58 (2009), pp. 579–588.
  • J. Wick, Numerical approaches to the kinetic semiconductor equation, Computing. 52 (1994), pp. 39–49.
  • Y. Chen. Model Order Reduction for Nonlinear Systems. M.S. Thesis, Massachusetts Institute of Technology, Cambridge, MA, September 1999.
  • L. Feng, Review of model order reduction methods for numerical simulation of nonlinear circuits, Appl. Math. Comput. 167 (2005), pp. 576–591.
  • M. Rewienski and J. White. Improving trajectory piecewise-linear approach to nonlinear model order reduction for micromachined devices using an aggregated projection basis, in Technical Proceedings of the 2002 International Conference on Modeling and Simulation of Microsystems, Chapter 3: System Level Modeling of MEMS, San Juan, Puerto Rico, 2002, pp. 128–131.
  • M. Rewienski. A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, June 2003.
  • J. Geiser and R. Roehle, Kinetic processes and phase-transition of CVD processes for Ti2SiC3, J. Convergence Inf. Technol. 5 (6) (2010), pp. 9–32.
  • J. Geiser, Computing exponential for iterative splitting methods: Algorithms and applications, J. Appl. Math. (2011), Article ID 193781.
  • G.M. Kepler, H.T. Tran, and H.T. Banks, Reduced order model compensator control of species transport in a CVD reactor, Optimal Control Appl. Methods. 21 (1999), pp. 143–160.
  • P.J. Antsaklis and A.N. Michel, Linear Systems, McGraw-Hill, New York, 1997.
  • L. Zadeh and C. Desoer, Linear System Theory. The State Space Approach, McGraw-Hill, New York, 1963.
  • B. Chapman, Glow Discharge Processes. Sputtering and Plasma Etching, John Wiley & Sons, New York, 1980.
  • N. Morosoff, Plasma Deposition, Treatment and Etching of Polymers, Academic Press, Boston, MA, 1990.
  • F. Langlais, F. Loumagne, D. Lespiaux, S. Schamm, and R. Naslain, Kinetic processes in the CVD of SiC from CH3SiCl3-H2 in a vertical hot-wall reactor, J. Phys. IV Colloque C5 Suppl J. Phys II. 5 (1995), pp. 105–115.
  • P. Favia, E. Sardella, R. Gristina, A. Millella, and R. D’Agostino, Functionalization of biomedical polymers by means of plasma processes: Plasma treated polymers with limited hydrophobic recovery and PE-CVD of COOH functional coatings, J. Polym. Sci. Technol. 15 (2) (2002), pp. 341–350.
  • H. Rouch. MOCVD Research Reactor Simulation. Proceedings of the COMSOL Users Conference 2006, Paris, France, 2006.
  • V.A. Kadetov. Diagnostics and modeling of an inductively coupled radio frequency discharge in hydrogen. Ph.D. thesis, Ruhr University of Bochum, Germany, 2004.
  • J. Geiser and M. Arab, Porous media-based modeling of PE-CVD apparatus: Electrical fields and deposition geometries, Spec. Topics Rev. Porous Media 1 (3) (2010), pp. 215–229.
  • L. Rudniak, Numerical simulation of chemical vapour deposition process in electric field, Comput. Chem. Eng. 22 (1998), pp. 755–758.
  • J. Geiser, Multiscale modeling of chemical vapor deposition (CVD) apparatus: Simulations and approximations, Spec. Issue Multiscale Simul. oft Matter. Polym. 5 (1) (2013), pp. 142–160.
  • A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed., Artech House, Norwood, MA, 2005.
  • P. Knabner and L. Angerman, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer-Verlag, New York, 2003.
  • J. Geiser, Iterative Splitting Methods for Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2011.
  • B. Gustafsson, High-Order Difference Methods for Time Dependent PDE, Springer-Verlag, Berlin, 2008.
  • C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, Series: Frontiers in Applied Mathematics, Vol. 16, SIAM, Philadelphia, PA, 1995.
  • J. Geiser, Iterative operator-splitting methods for nonlinear differential equations and applications, Numer. Methods Partial Differential Equations 27 (5) (2011), pp. 1026–1054.
  • A.D. Polyanin and V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed., Chapman & Hall, Boca Raton, FL, 2002.
  • S. Blanes, F. Casas, J.A. Oteo, and J. Ros, The Magnus expansion and some of its applications, Phys. Rep. 470 (5–6) (2009), pp. 151–238.
  • S. Blanes and P.C. Moan, Fourth- and sixth-order commutator-free Magnus integrators for linear and nonlinear dynamical systems, Appl. Numer. Math. 56 (2006), pp. 1519–1537.
  • F. Casas and A. Iserles, Explicit Magnus expansions for nonlinear equations, J. Phys. A Math. Gen. 39 (2006), pp. 5445–5461.
  • J. Geiser, Computing exponential for iterative splitting methods, J. Appl. Math. 2011 (2011), Article ID 193781.
  • D. Zwillinger (ed.), Bernoulli equation chapter II.A, 37, in Handbook of Differential Equations, 3rd ed., Academic Press, Boston, MA, 1997, pp. 157–158.
  • J. Geiser, Consistency of iterative operator-splitting method, Theory and Appl. published online. doi:10.1002/num. 20422
  • J. Geiser and R. Roehle, Kinetic processes and phase-transition of CVD processes for Ti2SiC3, J Convergence Inf. Technol. 5 (6) (2010), pp. 9–32.
  • R. Bellman, Introduction to Matrix Analysis, Mcgraw-Hill, New York, 1960.
  • J.S. Respondek, Approximate controllability of the nth order infinite dimensional systems with controls delayed by the control devices, Int. J. Syst. Sci. 39 (8) (2008), pp. 765–782.
  • I. Farago and J. Geiser. Iterative Operator-Splitting Methods for Linear Problems. Preprint No. 1043 of the Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany, June 2005.
  • U. Miekkala and O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems, SIAM J. Sci. Stat. Comput. 8 (1987), pp. 459–482.
  • S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems, B.G. Teubner, Stuttgart, 1993.

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