Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method.

*(English)*Zbl 1183.65152Summary: A Schur-decomposition for three-dimensional matrix equations is developed and used to directly solve the radiative discrete ordinates equations which are discretized by a Chebyshev collocation spectral method. Three methods, say, the spectral methods based on 2D and 3D matrix equation solvers individually, and the standard discrete ordinates method, are presented. The numerical results show the good accuracy of spectral method based on direct solvers. The CPU time cost comparisons against the resolutions between these three methods are made using MATLAB and FORTRAN 95 computer languages separately. The results show that the CPU time cost of Chebyshev collocation spectral method with 3D Schur-decomposition solver is the least, and almost only one thirtieth to one fiftieth CPU time is needed when using the spectral method with 3D Schur-decomposition solver compared with the standard discrete ordinates method.

##### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

##### Keywords:

Schur-decomposition; matrix-diagonalization; tensor product; spectral methods; radiative transfer equation; discrete ordinates method; Chebyshev collocation; numerical results
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\textit{B.-W. Li} et al., J. Comput. Phys. 229, No. 4, 1198--1212 (2010; Zbl 1183.65152)

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[1] | Golub, G.H.; Nash, S.; Van Loan, C.F., A hesenberg – schur method for the problem \(\mathit{AX} + \mathit{XB} = C\), IEEE trans. auto. control, 24, 909-913, (1979) · Zbl 0421.65022 |

[2] | Brierly, S.D.; Lee, E.B., Solution of the equation \(A(z) X(z) - X(z) B(z) = C(z)\) and its application to the stability of generalized linear system, Int. J. control, 40, 1065-1075, (1984) · Zbl 0546.93061 |

[3] | Lynch, R.E.; Rice, J.R.; Thomas, D.H., Direct solution of partial differential equations by tensor product method, Numer. math., 6, 185-199, (1964) · Zbl 0126.12703 |

[4] | Haidvogel, D.B.; Zang, T.A., The accurate solution of poisson’s equation by expansion in Chebyshev polynomials, J. comput. phys., 30, 167-180, (1979) · Zbl 0397.65077 |

[5] | Haldenwang, P.; Labrosse, G.; Abboude, S.; Deville, M.O., Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equation, J. comput. phys., 55, 115-128, (1984) · Zbl 0544.65071 |

[6] | Ehrenstein, U.; Peyret, R., A Chebyshev collocation method for the navier – stokes equations with application to double-diffusive convection, Int. J. numer. methods fluids, 9, 427-452, (1989) · Zbl 0665.76107 |

[7] | Chen, H.-L.; Su, Y.-H.; Shizgal, B.D., A direct spectral collocation Poisson solver in polar and cylindrical coordinates, J. comput. phys., 160, 453-469, (2000) · Zbl 0951.65125 |

[8] | Bartels, R.H.; Stewart, G.W., Solution of the matrix equation \(\mathit{AX} + \mathit{XB} = C\), Commun. ACM, 15, 820-826, (1972) · Zbl 1372.65121 |

[9] | Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1991), Cambridge University Press Cambridge · Zbl 0729.15001 |

[10] | Hoskins, W.D.; Meek, D.S.; Walton, D.J., The numerical solution of the matrix equation \(\mathit{XA} + \mathit{AY} = F\), BIT numer. math., 17, 184-190, (1977) · Zbl 0358.65025 |

[11] | Peyret, R., Spectral methods for incompressible viscous flow, (2002), Springer-Verlage New York · Zbl 1005.76001 |

[12] | R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, 4th ed., Taylor and Francis, Washington, DC. |

[13] | D. Gottlieb, S.A. Orszag, Numerical analysis of spectral methods: Theory and applications, Regional Conference Series in Applied Mathematics, vol. 28, Philadelphia, SIAM, 1977. · Zbl 0412.65058 |

[14] | Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1989), Springer-Verlag Berlin |

[15] | Li, B.-W.; Sun, Y.-S.; Yu, Y., Iterative and direct Chebyshev collocation spectral methods for one-dimensional radiative heat transfer, Int. J. heat mass transfer, 51, 5887-5894, (2008) · Zbl 1153.80323 |

[16] | Sun, Y.-S.; Li, B.-W., Chebyshev collocation spectral method for one-dimensional radiative heat transfer in graded index media, Int. J. thermal sci., 48, 691-698, (2009) |

[17] | Li, B.-W.; Sun, Y.-S.; Zhang, D.-W., Chebyshev collocation spectral methods for coupled radiation and conduction in a concentric spherical participating medium, J. heat transfer, 131, 062701-062709, (2009) |

[18] | Y.-S. Sun, B.-W. Li, Chebyshev collocation spectral approach for combined radiative and conduction heat transfer in one-dimensional semitransparent medium with graded index, Int. J. Heat Mass Transfer (2009), doi:10.1115/1.4000444. |

[19] | Y.-S. Sun, B.-W. Li, Spectral collocation method for transient combined radiation and conduction in an anisotropic scattering slab with graded index, J. Heat Transfer, in press. |

[20] | Selcuk, N., Exact solutions for radiative heat transfer in box-shaped furnaces, J. heat transfer, 107, 648-655, (1985) |

[21] | Carlson, B.G.; Lathrop, K.D., Computing methods in reactor physics, () |

[22] | Fiveland, W.A., Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures, J. heat transfer, 106, 699-706, (1984) |

[23] | Truelove, J.S., Discrete-ordinates solutions of radiation transport equation, J. heat transfer, 109, 1048-1051, (1987) |

[24] | Truelove, J.S., Three-dimensional radiation in absorbing-emitting-scattering media using the discrete-ordinates approximation, Jqsrt, 39, 27-31, (1988) |

[25] | Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods/fundamentals in single domains, (2006), Springer-Verlag Berlin · Zbl 1093.76002 |

[26] | Trefethen, L.N., Spectral methods in MATLAB, (2000), SIAM Oxford University · Zbl 0953.68643 |

[27] | Henshall, P.; Palmer, P., A leapfrog algorithm for coupled conductive and radiative transient heat transfer in participating media, Int. J. thermal sci., 47, 388-398, (2008) |

[28] | Selcuk, N.; Doner, N., A 3-D radiation model for non-grey gases, Jqsrt, 110, 184-191, (2009) |

[29] | Edström, P., Numerical performance of stability enhancing and speed increasing steps in radiative transfer solution methods, J. comput. appl. math., 228, 104-114, (2009) · Zbl 1165.65405 |

[30] | Selcuk, N.; Kayakol, N., Evaluation of discrete ordinates method for radiative transfer in rectangular furnaces, Int. J. heat mass transfer, 40, 213-222, (1997) · Zbl 0925.76410 |

[31] | Li, B.-W.; Chen, H.-G.; Zhou, J.-H.; Cao, X.-Y.; Cen, K.-F., The spherical surface symmetrical equal dividing angular quadrature scheme for discrete ordinates method, J. heat transfer, 124, 482-489, (2002) |

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