# Bicompact schemes for an inhomogeneous linear transport equation

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## Abstract

The bicompact finite-difference schemes constructed for a homogeneous linear transport equation for the case of the inhomogeneous transport equation are generalized. The equation describes the transport of particles or radiation in media. Using the method of lines, the bicompact scheme is constructed for the initial unknown function and the complementary unknown mesh function defined as the integral average of the initial function with respect to space cells. The comparison of the calculation results of the proposed method and the conservative-characteristic method is carried out. The latter can be assigned to the class of bicompact finite-difference schemes; however, this method is based on the idea of the redistribution of incoming fluxes from illuminated edges to unilluminated edges.

### Keywords

transport equation finite-difference schemes bicompact schemes conservative schemes Runge-Kutta methods redistribution of fluxes## Preview

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### References

- 1.B. V. Rogov and M. N. Mikhailovskaya, “Fourth-order accurate bicompact schemes for hyperbolic equations,” Dokl. Math.
**81**(1), 146–150 (2010).MathSciNetCrossRefMATHGoogle Scholar - 2.B. V. Rogov and M. N. Mikhailovskaya, “Monotonic bicompact schemes for linear transport equations,” Mat. Models Comput. Simul.
**4**(1), 92–100 (2012).MathSciNetCrossRefGoogle Scholar - 3.B. V. Rogov and M. N. Mikhailovskaya, “Monotone bicompact schemes for a linear advection equation,” Dokl. Math.
**83**(1), 121–125 (2011).MathSciNetCrossRefMATHGoogle Scholar - 4.B. V. Rogov and M. N. Mikhailovskaya, “Monotone high-order accurate compact scheme for quasilinear hyperbolic equations,” Dokl. Math.
**84**(2), 447–452 (2011).MathSciNetCrossRefGoogle Scholar - 5.M. N. Mikhailovskaya and B. V. Rogov, “Bicompact monotonic schemes for a multidimensional linear transport equation,” Mat. Models Comput. Simul.
**4**(3), 355–362 (2012).MathSciNetCrossRefGoogle Scholar - 6.M. N. Mikhailovskaya and B. V. Rogov, “Monotone compact running schemes for systems of hyperbolic equations,” Comp. Math. Math. Phys.
**52**(4), 578–600 (2012).MathSciNetCrossRefGoogle Scholar - 7.E. N. Aristova, D. F. Baidin, and V. Ya. Gol’din, “Two variants of the efficient method to solve a transport equation in r-z geometry based on the transition to Vladimirov variables,” Mat. Model.
**18**(7), 43–52 (2006).MATHGoogle Scholar - 8.A. Harten, “Class of high resolution total variation stable finite-difference schemes,” SIAM J. Numer. Anal.
**21**, 1–23 (1984).MathSciNetCrossRefMATHGoogle Scholar - 9.A. Harten, “ENO schemes with subcell resolution,” J. Comp. Phys.
**83**, 148–184 (1989).MathSciNetCrossRefMATHGoogle Scholar - 10.K. A. Mathews, “On the propagation of rays in discrete ordinates,” Nucl. Sci. Eng.
**123**, 155–180 (1999).Google Scholar - 11.K. D. Lathrop, “Spatial differencing of the transport equation: positivity vs. accuracy,” J. Comput. Phys.
**4**(4), 475–490 (1969).CrossRefMATHGoogle Scholar - 12.L. P. Bass and O. V. Nikolaeva, “SWDD OAM scheme and results of methodical calculations” in
*Neitronika-97: Algorithms and Programs for Neutronic Calculations of Nuclear Reactors*(Obninsk, 1997), pp. 76–83 [in Russian].Google Scholar - 13.O. V. Nikolaeva, “Special grid approximations for the transport equation in strongly heterogeneous vedia with the (
*x*,*y*)-geometry,” Comput. Math. Math. Phys.**44**(5), 835–855 (2004).MathSciNetGoogle Scholar - 14.A. M. Voloshchenko, “Solution of a transfer equation by DSn method in heterogeneous environments II: Onedimensional spherical and cylindrical geometries” in
*Numerical Solution of Transfer Equations in One-Dimensional Problems*(Keldysh Institute of Appl. Math., Moscow, 1981), pp. 64–91 [in Russian].Google Scholar - 15.K. Dekker and J. Verwer,
*Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations*(North-Holland Elsevier Science Publishers, Amsterdam, 1984).MATHGoogle Scholar - 16.E. Hairer and G. Wanner,
*Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems*(Springer, Berlin, 1996).CrossRefMATHGoogle Scholar - 17.B. V. Rogov and M. N. Mikhailovskaya, “On the convergence of compact difference schemes,” Mat. Models Comput. Simul.
**1**(1), 91–104 (2009).MathSciNetCrossRefGoogle Scholar - 18.E. N. Aristova and B. V. Rogov, “Boundary conditions implementation in bicompact schemes for the linear transport equation,” Mat. Models Comput. Simul.
**5**(3), 199–207 (2013).CrossRefMathSciNetGoogle Scholar - 19.V. Ya. Gol’din, G. V. Danilova, and N. N. Kalitkin, “Numerical integration of a multidimensional transfer equation” in
*Numerical Methods for Solving Problems of Mathematical Physics*(Nauka, Moscow, 1966), pp. 190–193 [in Russian].Google Scholar - 20.V. Ya. Gol’din, N. N. Kalitkin, and T. V. Shishova, “Nonlinear difference schemes for hyperbolic equations,” Zh. Vychisl. Mat. Mat. Fiz.
**5**(5), 938–944 (1965).Google Scholar - 21.M. I. Bakirova, V. Ya. Karpov, and M. I. Mukhina, “Characteristic and interpolation method for solving a transfer equation,” Differ. Uravn.
**22**(7), 1141–1148 (1986).MathSciNetGoogle Scholar - 22.V. E. Troshchiev, A. V. Nifanova, and Yu. V. Troshchiev, “Characteristic approach to the approximation of conservation laws in radiation transfer kinetic equations,” Dokl. Math.
**69**(1), 136–140 (2004).MATHGoogle Scholar