Abstract
The method for constructing upwind high-resolution schemes is proposed in application to the modeling of ionizing waves in gas discharges. The flux-limiting criterion for continuity equations is derived using the proposed partial monotony property of a finite difference scheme. For two-dimensional extension, the cone transport upwind approach for constructing genuinely two-dimensional difference schemes is used. It is shown that when calculating rotations of symmetric profiles by using this scheme, a circular form of isolines is not distorted in a distinct from the coordinate splitting method. The conservative second order finite-difference scheme is proposed for solving the equations system of the drift-diffusion model of electric discharge; this scheme implies finite-difference conservation laws of electric charge and full electric current (fully conservative scheme). Computations demonstrate absence of numeric oscillations and good resolution of two-dimensional ionizing fronts in simulations of streamer and barrier discharges
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References
A. F. Diyakov, Yu. K. Bobrov, A. V. Sorokin, and Yu. V. Yurgelenas, Physics Fundamentals of Electric Breakdown in Gases (MPEI Publishers, Moscow, 1999) [in Russian].
A. J. Davies, “Discharge Simulation,” Proc. IEE, Part A: Sci. Meas. Technol. 133(4), 217–240 (1986).
R. Morrow and L. E. Cram, “Flux-Corrected Transport and Diffusion on a Non-Uniform Mesh,” J. Comput. Phys. 57(1), 129–136 (1985).
D. L. Book and J. P. Boris, “Flux-Corrected Transport, II: Generalizations of the Method,” J. Comput. Phys. 18(3), 248–283 (1975).
B. van Leer, “Towards the Ultimate Conservative Difference Scheme, IV: A New Approach to Numerical Convection,” J. Comput. Phys. 23(2), 276–299 (1977).
A. Harten, “High Resolution Schemes for Hyperbolic Conservation Laws,” J. Comput. Phys. 49, 357–393 (1983).
P. K. Sweby, “High Resolution TVD Schemes Using Flux Limiters,” Lect. Appl. Math. 22, 289–309 (1985).
B. P. Leonard, “The ULTIMATE Conservative Difference Scheme Applied to Unsteady One-Dimensional Advection,” Comput. Meth. Appl. Mech. Engng. 88(1), 17–74 (1991).
C. D. Munz, “On the Numerical Dissipation of High Resolution Schemes for Hyperbolic Conservation Laws,” J. Comput. Phys. 77(1), 18–39 (1988).
E. E. Kunhardt and C.-H. Wu, “Towards a More Accurate Flux-Corrected Transport Algorithm,” J. Comput. Phys. 68(1), 127–150 (1987).
Yu. K. Bobrov and Yu. V. Yurghelenas, “Application of High-Resolution Schemes in the Modeling of Ionization Waves in Gas Discharge,” Comput. Math. Math. Phys. 38(10), 1652–1661 (1998).
P. Colella, “Multidimensional Upwind Methods for Hyperbolic Conservation Laws,” J. Comput. Phys. 87, 171–200 (1990).
B. van Leer, “Multidimensional Explicit Difference Schemes for Hyperbolic Conservation Laws,” in Comput. Meth. Appl. Sci. Engng (Elsevier Sci. Publ. B.V., Amsterdam, 1994).
Yu. V. Yourguelenas, “Fully Conservative Monotonic Numerical Scheme for Streamer Plasma 2D-Modelling,” Czech. J. Phys. 50(Suppl. S3), 301–304 (2000).
Yu. V. Yourguelenas, “Monotonic Fully Conservative Numerical Scheme for Streamer Discharge Simulation,” in 7th Internat. Symp. High Pressure Low Temperature Plasma Chemistry (HAKONE VII), Greiswald, Germany, 2000, Vol. 1, pp. 159–163.
H.-E. Wagner, Yu. V. Yurgelenas, and R. Brandenburg, “The Development of Microdischarges of Barrier Discharges in N2/O2 Mixtures—Experimental Investigations and Modelling,” Plasma Phys. Control. Fusion 47, B641–B654 (2005).
Yu. V. Yurgelenas and H.-E. Wagner, “A Computational Model of a Barrier Discharge in Air at Atmospheric Pressure: the Role of Residual Surface Charges in Microdischarge Formation,” J. Phys. D: Appl. Phys. 39, 4031–4043 (2006).
Y. V. Yurgelenas and M. A. Leeva, “Development of a Barrier Discharge in Air in Highly Nonhomogeneous Electric Field Caused by the Residual Dielectric Surface Charges,” IEEE Trans. Plasma Sci. 37, 809–815 (2009).
P. Lax and B. Wendroff, “Systems of Conservation Laws,” Communs Pure Appl. Math. 13, 217–237 (1960).
R. F. Warming and R. M. Beam, “Upwind Second Order Difference Schemes and Application in Aerodynamics,” AIAA J. 14, 1241–1249 (1976).
J. E. Fromm, “A Method of Reducing Dispersion in Convective Difference Schemes,” J. Comput. Phys. 3, 176–189 (1968).
S. A. Velichko, Yu. B. Lifshitz, and I. A. Solntsev, High Accurate Scheme for Flow Computation over Moving Body, Preprint No. 92 (TsAGI, Moscow, 1996) [in Russian].
Yu. B. Radvogin, Quasi-Monotonic Multidimensional Difference Schemes of Second-Order Accuracy, Preprint No. 19 (Keldysh Inst. Appl. Math. RAS, Moscow, 1991) [in Russian].
H. T. Huynh, Accurate Upwind Schemes for the Euler Equations: AIAA paper 95-1737, pp. 1022–1039, 1995.
A. A. Kulikovsky, “Two-Dimensional Simulation of Positive Streamer in N2 between Parallel-Plate Electrodes,” J. Phys. D: Appl. Phys. 28, 2483–2493 (1995).
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Published in Russian in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 8, pp. 1420–1437
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Yurgelenas, Y.V. Two-dimensional high-resolution schemes and their application in the modeling of ionizing waves in gas discharges. Comput. Math. and Math. Phys. 50, 1350–1366 (2010). https://doi.org/10.1134/S0965542510080075
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DOI: https://doi.org/10.1134/S0965542510080075