Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media

Knut Stamnes; S-Chee Tsay; Warren Wiscombe; Kolf Jayaweera

doi:10.1364/AO.27.002502

Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media

Knut Stamnes, S-Chee Tsay, Warren Wiscombe, and Kolf Jayaweera

Applied Optics
Vol. 27,
Issue 12,
pp. 2502-2509
(1988)
•https://doi.org/10.1364/AO.27.002502

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Knut Stamnes, S-Chee Tsay, Warren Wiscombe, and Kolf Jayaweera, "Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media," Appl. Opt. 27, 2502-2509 (1988)

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Abstract

We summarize an advanced, thoroughly documented, and quite general purpose discrete ordinate algorithm for time-independent transfer calculations in vertically inhomogeneous, nonisothermal, plane-parallel media. Atmospheric applications ranging from the UV to the radar region of the electromagnetic spectrum are possible. The physical processes included are thermal emission, scattering, absorption, and bidirectional reflection and emissionat the lower boundary. The medium may be forced at the top boundary by parallel or diffuse radiation and by internal and boundary thermal sources as well. We provide a brief account of the theoretical basis as well as a discussion of the numerical implementation of the theory. The recent advances made by ourselves and our collaborators—advances in both formulation and numerical solution—are all incorporated in the algorithm. Prominent among these advances are the complete conquest of two ill-conditioning problems which afflicted all previous discrete ordinate implementations: (1) the computation of eigenvalues and eigenvectors and (2) the inversion of the matrix determining the constants of integration. Copies of the fortran program on microcomputer diskettes are available for interested users.

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References

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Publication

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
K.-N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmosphere,” J. Atmos. Sci. 30, 1303 (1973).
[Crossref]
K. Stamnes, R. A. Swanson, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculations in Anisotropically Scattering Atmospheres,” J. Atmos. Sci. 38, 387 (1981).
[Crossref]
K. Stamnes, H. Dale, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculation in Anisotropically Scattering Atmospheres, II: Intensity Computations,” J. Atmos. Sci. 38, 2696 (1981).
[Crossref]
K. Stamnes, “On the Computation of Angular Distributions of Radiation in Planetary Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 28, 47 (1982a).
[Crossref]
K. Stamnes, “Reflection and Transmission by a Vertically Inhomogeneous Planetary Atmosphere,” Planet. Space Sci. 30, 727 (1982b).
[Crossref]
K. Stamnes, P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 31, 273 (1984).
[Crossref]
K. Stamnes, S.-C. Tsay, T. Nakajima, “Computation of Eigenvaluesand Eigenvectors for Discrete Ordinate and Matrix Operator Method Radiative Transfer,” J. Quant. Spectrosc. Radiat. Transfer in press (1988).
[Crossref]
K. Stamnes, S.-C. Tsay, W. J. Wiscombe, K. Jayaweera, “An Improved, Numerically Stable Computer Code for Discrete-Ordinate-Method Radiative Transfer in Scattering and Emitting Layered Meda,” NASA Report, in press (1988).
K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Orlando, FL, 1980).
J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974).
[Crossref]
W. M. Irvine, “Multiple Scattering in Planetary Atmospheres,” Icarus 25, 175 (1975).
[Crossref]
J. Lenoble, Ed., Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (A. Deepak, Hampton, VA, 1985).
W. J. Wiscombe, “Atmospheric Radiation: 1975–1983,” Rev. Geophys. 21, 957 (1983).
[Crossref]
K. Stamnes, “The Theory of Multiple Scattering of Radiation in Plane Parallel Atmospheres,” Rev. Geophys. 24, 299 (1986).
[Crossref]
T. Nakajima, M. Tanaka, “Matrix Formulations for the Transfer of Solar Radiation in a Plane-Parallel Atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 35, 13 (1986).
[Crossref]
W. J. Wiscombe, “The Delta-M Method: Rapid Yet Accurate Radiative Flux Calculations for Strongly Asymmetric Phase Functions,” J. Atmos. Sci. 34, 1408 (1977).
[Crossref]
J. B. Sykes, “Approximate Integration of the Equation of Transfer,” Mon. Not. Roy. Astron. Soc. 11, 377 (1951).
W. J. Wiscombe, “Extension of the Doubling Method to Inhomogeneous Sources,” J. Quant. Spectrosc. Radiat. Transfer 16, 477 (1976).
[Crossref]
W. R. Cowell, Ed., Sources and Developments of Mathematical Software (Prentice Hall, Englewood Cliffs, NJ, 1980).
J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, linpack User’s Guide (SIAM, Philadelphia, 1979).
V. Kourganoff, Basic Methods in Transfer Problems (Dover, New York, 1963).

1986 (2)

K. Stamnes, “The Theory of Multiple Scattering of Radiation in Plane Parallel Atmospheres,” Rev. Geophys. 24, 299 (1986).
[Crossref]

T. Nakajima, M. Tanaka, “Matrix Formulations for the Transfer of Solar Radiation in a Plane-Parallel Atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 35, 13 (1986).
[Crossref]

1984 (1)

K. Stamnes, P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 31, 273 (1984).
[Crossref]

1983 (1)

W. J. Wiscombe, “Atmospheric Radiation: 1975–1983,” Rev. Geophys. 21, 957 (1983).
[Crossref]

1982 (2)

K. Stamnes, “On the Computation of Angular Distributions of Radiation in Planetary Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 28, 47 (1982a).
[Crossref]

K. Stamnes, “Reflection and Transmission by a Vertically Inhomogeneous Planetary Atmosphere,” Planet. Space Sci. 30, 727 (1982b).
[Crossref]

1981 (2)

K. Stamnes, R. A. Swanson, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculations in Anisotropically Scattering Atmospheres,” J. Atmos. Sci. 38, 387 (1981).
[Crossref]

K. Stamnes, H. Dale, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculation in Anisotropically Scattering Atmospheres, II: Intensity Computations,” J. Atmos. Sci. 38, 2696 (1981).
[Crossref]

1977 (1)

W. J. Wiscombe, “The Delta-M Method: Rapid Yet Accurate Radiative Flux Calculations for Strongly Asymmetric Phase Functions,” J. Atmos. Sci. 34, 1408 (1977).
[Crossref]

1976 (1)

W. J. Wiscombe, “Extension of the Doubling Method to Inhomogeneous Sources,” J. Quant. Spectrosc. Radiat. Transfer 16, 477 (1976).
[Crossref]

1975 (1)

W. M. Irvine, “Multiple Scattering in Planetary Atmospheres,” Icarus 25, 175 (1975).
[Crossref]

1974 (1)

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974).
[Crossref]

1973 (1)

K.-N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmosphere,” J. Atmos. Sci. 30, 1303 (1973).
[Crossref]

1951 (1)

J. B. Sykes, “Approximate Integration of the Equation of Transfer,” Mon. Not. Roy. Astron. Soc. 11, 377 (1951).

Bunch, J. R.

J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, linpack User’s Guide (SIAM, Philadelphia, 1979).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Conklin, P.

Dale, H.

Dongarra, J. J.

J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, linpack User’s Guide (SIAM, Philadelphia, 1979).

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974).
[Crossref]

Irvine, W. M.

W. M. Irvine, “Multiple Scattering in Planetary Atmospheres,” Icarus 25, 175 (1975).
[Crossref]

Jayaweera, K.

K. Stamnes, S.-C. Tsay, W. J. Wiscombe, K. Jayaweera, “An Improved, Numerically Stable Computer Code for Discrete-Ordinate-Method Radiative Transfer in Scattering and Emitting Layered Meda,” NASA Report, in press (1988).

Kourganoff, V.

V. Kourganoff, Basic Methods in Transfer Problems (Dover, New York, 1963).

Liou, K. N.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Orlando, FL, 1980).

Liou, K.-N.

K.-N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmosphere,” J. Atmos. Sci. 30, 1303 (1973).
[Crossref]

Moler, C. B.

J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, linpack User’s Guide (SIAM, Philadelphia, 1979).

Nakajima, T.

T. Nakajima, M. Tanaka, “Matrix Formulations for the Transfer of Solar Radiation in a Plane-Parallel Atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 35, 13 (1986).
[Crossref]

K. Stamnes, S.-C. Tsay, T. Nakajima, “Computation of Eigenvaluesand Eigenvectors for Discrete Ordinate and Matrix Operator Method Radiative Transfer,” J. Quant. Spectrosc. Radiat. Transfer in press (1988).
[Crossref]

Stamnes, K.

K. Stamnes, “The Theory of Multiple Scattering of Radiation in Plane Parallel Atmospheres,” Rev. Geophys. 24, 299 (1986).
[Crossref]

K. Stamnes, “On the Computation of Angular Distributions of Radiation in Planetary Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 28, 47 (1982a).
[Crossref]

K. Stamnes, “Reflection and Transmission by a Vertically Inhomogeneous Planetary Atmosphere,” Planet. Space Sci. 30, 727 (1982b).
[Crossref]

K. Stamnes, R. A. Swanson, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculations in Anisotropically Scattering Atmospheres,” J. Atmos. Sci. 38, 387 (1981).
[Crossref]

Stewart, G. W.

J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, linpack User’s Guide (SIAM, Philadelphia, 1979).

Swanson, R. A.

K. Stamnes, R. A. Swanson, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculations in Anisotropically Scattering Atmospheres,” J. Atmos. Sci. 38, 387 (1981).
[Crossref]

Sykes, J. B.

J. B. Sykes, “Approximate Integration of the Equation of Transfer,” Mon. Not. Roy. Astron. Soc. 11, 377 (1951).

Tanaka, M.

T. Nakajima, M. Tanaka, “Matrix Formulations for the Transfer of Solar Radiation in a Plane-Parallel Atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 35, 13 (1986).
[Crossref]

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974).
[Crossref]

Tsay, S.-C.

Wiscombe, W. J.

W. J. Wiscombe, “Atmospheric Radiation: 1975–1983,” Rev. Geophys. 21, 957 (1983).
[Crossref]

W. J. Wiscombe, “The Delta-M Method: Rapid Yet Accurate Radiative Flux Calculations for Strongly Asymmetric Phase Functions,” J. Atmos. Sci. 34, 1408 (1977).
[Crossref]

W. J. Wiscombe, “Extension of the Doubling Method to Inhomogeneous Sources,” J. Quant. Spectrosc. Radiat. Transfer 16, 477 (1976).
[Crossref]

Icarus (1)

W. M. Irvine, “Multiple Scattering in Planetary Atmospheres,” Icarus 25, 175 (1975).
[Crossref]

J. Atmos. Sci. (4)

W. J. Wiscombe, “The Delta-M Method: Rapid Yet Accurate Radiative Flux Calculations for Strongly Asymmetric Phase Functions,” J. Atmos. Sci. 34, 1408 (1977).
[Crossref]

K.-N. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete-Ordinate Method for Radiative Transfer: Applications to Cloudy and Hazy Atmosphere,” J. Atmos. Sci. 30, 1303 (1973).
[Crossref]

K. Stamnes, R. A. Swanson, “A New Look at the Discrete Ordinate Method for Radiative Transfer Calculations in Anisotropically Scattering Atmospheres,” J. Atmos. Sci. 38, 387 (1981).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (4)

K. Stamnes, “On the Computation of Angular Distributions of Radiation in Planetary Atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 28, 47 (1982a).
[Crossref]

W. J. Wiscombe, “Extension of the Doubling Method to Inhomogeneous Sources,” J. Quant. Spectrosc. Radiat. Transfer 16, 477 (1976).
[Crossref]

T. Nakajima, M. Tanaka, “Matrix Formulations for the Transfer of Solar Radiation in a Plane-Parallel Atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 35, 13 (1986).
[Crossref]

Mon. Not. Roy. Astron. Soc. (1)

J. B. Sykes, “Approximate Integration of the Equation of Transfer,” Mon. Not. Roy. Astron. Soc. 11, 377 (1951).

Planet. Space Sci. (1)

K. Stamnes, “Reflection and Transmission by a Vertically Inhomogeneous Planetary Atmosphere,” Planet. Space Sci. 30, 727 (1982b).
[Crossref]

Rev. Geophys. (2)

W. J. Wiscombe, “Atmospheric Radiation: 1975–1983,” Rev. Geophys. 21, 957 (1983).
[Crossref]

K. Stamnes, “The Theory of Multiple Scattering of Radiation in Plane Parallel Atmospheres,” Rev. Geophys. 24, 299 (1986).
[Crossref]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974).
[Crossref]

Other (8)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

J. Lenoble, Ed., Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (A. Deepak, Hampton, VA, 1985).

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Orlando, FL, 1980).

W. R. Cowell, Ed., Sources and Developments of Mathematical Software (Prentice Hall, Englewood Cliffs, NJ, 1980).

J. J. Dongarra, C. B. Moler, J. R. Bunch, G. W. Stewart, linpack User’s Guide (SIAM, Philadelphia, 1979).

V. Kourganoff, Basic Methods in Transfer Problems (Dover, New York, 1963).

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Figures (1)

Fig. 1

Fig. 1 Schematic illustration of a multilayered medium.

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Equations (71)

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μ \frac{d u_{ν} (τ_{ν}, μ, ϕ)}{d τ} = u_{ν} (τ_{ν}, μ, ϕ) - S_{ν} (τ_{ν}, μ, ϕ),

S_{ν} (τ_{ν}, μ, ϕ) = \frac{ω_{ν} (τ_{ν})}{4 π} \int_{0}^{2 π} d ϕ^{'} \int_{- 1}^{1} d μ^{'} P_{ν} (τ_{ν}, μ, ϕ; μ^{'}, ϕ^{'}) \times u_{ν} (τ_{ν}, μ^{'}, ϕ^{'}) + Q_{ν} (τ_{ν}, μ, ϕ),

Q_{ν}^{(thermal)} (τ_{ν}) = [1 - ω_{ν} (τ_{ν})] B_{ν} [T (τ_{ν})],

Q_{ν}^{(beam)} (τ_{ν}, μ, ϕ) = \frac{ω_{ν} (τ_{ν}) I_{0}}{4 π} P_{ν} (τ_{ν}, μ, ϕ; - μ_{0}, ϕ_{0}) exp (- τ_{ν} / μ_{0}),

Q_{ν} (τ_{ν}, μ, ϕ) = Q_{ν}^{(thermal)} (τ_{ν}) + Q_{ν}^{(beam)} (τ_{ν}, μ, ϕ) .

u (τ, μ, ϕ) = \sum_{m = 0}^{2 N - 1} u^{m} (τ, μ) cos m (ϕ_{0} - ϕ),

μ \frac{d u^{m} (τ, μ)}{d τ} = u^{m} (τ, μ) - \int_{- 1}^{1} D^{m} (τ, μ, μ^{'}) u^{m} (τ, μ^{'}) d μ^{'} - Q^{m} (τ, μ) (m = 0, 1, 2, \dots, 2 N - 1),

D^{m} (τ, μ, μ^{'}) = \frac{ω (τ)}{2} \sum_{l = m}^{2 N - 1} (2 l + 1) g_{l}^{m} (τ) P_{l}^{m} (μ) P_{l}^{m} (μ^{'})

Q^{m} (τ, μ) = X_{0}^{m} (τ, μ) exp (- τ / μ_{0}) + δ_{m 0} Q^{(thermal)} (τ),

X_{0}^{m} (τ, μ) = \frac{ω (τ) I_{0}}{4 π} (2 - δ_{m 0}) \sum_{l = 0}^{2 N - 1} {(- 1)}^{l + m} (2 l + 1) \times g_{l}^{m} (τ) P_{l}^{m} (μ) P_{l}^{m} (μ_{0}),

\begin{array}{l} δ_{m 0} = 1 if m = 0 (0 otherwise), g_{l}^{m} (τ) = g_{l} (τ) \frac{(l - m)!}{(l + m)!}, \\ g_{l} (τ) = \frac{1}{2} \int_{- 1}^{1} P_{l} (cos θ) P (τ, cos θ) d cos θ . \end{array}

μ_{i} \frac{d u^{m} (τ, μ_{i})}{d τ} = u^{m} (τ, μ_{i}) - \sum_{\begin{array}{l} j = - N \\ j \neq 0 \end{array}}^{N} w_{j} D^{m} (τ, μ_{i}, μ_{j}) \times u^{m} (τ, μ_{j}) - Q^{m} (τ, μ_{i}) (i = \pm 1, \dots \pm N),

| \begin{matrix} \frac{d u^{+}}{d τ} \\ \frac{d u^{-}}{d τ} \end{matrix} | = | \begin{array}{r} - α & - β \\ β & α \end{array} | | \begin{matrix} u^{+} \\ u^{-} \end{matrix} | + | \begin{matrix} {\tilde{Q}}^{+} \\ {\tilde{Q}}^{-} \end{matrix} |,

\begin{array}{l} u^{\pm} = [u^{m} (τ, \pm μ_{i})] i = 1, \dots, N, \\ {\tilde{Q}}^{\pm} = M^{- 1} Q^{\pm}, \\ Q^{\pm} = Q^{m} (τ, \pm μ_{i}) i = 1, \dots, N, \\ M = {μ_{i} δ_{i j}} i, j = 1, \dots, N, \\ α = M^{- 1} (D^{+} W - I), \\ β = M^{- 1} D^{-} W, \\ W = (w_{i} δ_{i j}) i, j = 1, \dots, N, \\ D^{+} = [D^{m} (μ_{i}, μ_{j})] = [D^{m} (- μ_{i}, - μ_{j})] i, j = 1, \dots, N, \\ D^{-} = [D^{m} (- μ_{i}, μ_{j})] = [D^{m} (μ_{i}, - μ_{j})] i, j = 1, \dots, N . \end{array}

\sum_{\begin{array}{l} j = - N \\ j \neq 0 \end{array}}^{N} w_{j} D^{0} (τ, μ_{i}, μ_{j}) = \sum_{\begin{array}{l} i = - N \\ i \neq 0 \end{array}}^{N} w_{i} D^{0} (τ, μ_{i}, μ_{j}) = ω (τ),

u^{\pm} = G^{\pm} exp (- k τ),

| \begin{array}{r} α & β \\ - β & - α \end{array} | | \begin{matrix} G^{+} \\ G^{-} \end{matrix} | = k | \begin{matrix} G^{+} \\ G^{-} \end{matrix} |,

\begin{array}{l} α G^{+} + β G^{-} = k G^{+}, \\ β G^{+} + α G^{-} = - k G^{-}, \end{array}

(α - β) (G^{+} - G^{-}) = k (G^{+} + G^{-}),

(α + β) (G^{+} + G^{-}) = k (G^{+} - G^{-}) .

(α - β) (α + β) (G^{+} + G^{-}) = k^{2} (G^{+} + G^{-}),

u (τ, μ_{i}) = Z_{0} (μ_{i}) exp (- τ / μ_{0}) .

B [T (τ)] = \sum_{l = 0}^{K} b_{l} τ^{l},

u (τ, μ_{i}) = \sum_{l = 0}^{K} Y_{l} (μ_{i}) τ^{l} .

u (τ, μ_{i}) = \sum_{j = - N}^{N} C_{j} G_{j} (μ_{i}) exp (- k_{j} τ) + δ_{m 0} \sum_{l = 0}^{N} Y_{l} (μ_{i}) τ^{l} .

C_{0} G_{0} (μ_{i}) \equiv Z_{0} (μ_{i}),

k_{0} \equiv 1 / μ_{0} .

u (τ = 0, - μ, ϕ) = u_{\infty} (μ, ϕ),

u (τ = τ_{L}, + μ, ϕ) = u_{g} (μ, ϕ),

u (τ_{L}, + μ, ϕ) = ∊ (μ) B (T_{g}) + \frac{1}{π} \int_{0}^{2 π} d ϕ^{'} \int_{0}^{1} ρ_{d} (μ, ϕ; - μ^{'}, ϕ^{'}) u (τ_{L}, - μ^{'}, ϕ^{'}) μ^{'} d μ^{'} + \frac{μ_{0}}{π} I_{0} exp (- τ_{L} / μ_{0}) ρ_{d} (μ, ϕ; - μ_{0}, ϕ_{0}),

∊ (μ) + \frac{1}{π} \int_{0}^{2 π} d ϕ^{'} \int_{0}^{1} ρ_{d} (μ, ϕ; - μ^{'}, ϕ^{'}) μ^{'} d μ^{'} = 1

ρ_{d} (μ, ϕ; - μ^{'}, ϕ^{'}) = \sum_{l = 0}^{2 N - 1} (2 l + 1) h_{l} P_{l} (cos θ),

cos θ = - μ μ^{'} + {(1 - μ^{2})}^{1 / 2} {(1 - μ^{2})}^{1 / 2} cos (ϕ^{'} - ϕ) .

ρ_{d} (μ, ϕ; - μ^{'}, ϕ^{'}) = \sum_{m = 0}^{2 N - 1} ρ_{d}^{m} (μ, - μ^{'}) cos m (ϕ^{'} - ϕ),

ρ_{d}^{m} (μ, - μ^{'}) = (2 - δ_{0 m}) \sum_{l = m}^{2 N - 1} (2 l + 1) h_{l}^{m} P_{l}^{m} (- μ^{'}) P_{l}^{m} (μ),

\begin{array}{l} h_{l}^{m} = h_{l} \frac{(l - m)!}{(l + m)!}, \\ h_{l} = \frac{1}{2} \int_{- 1}^{1} P_{l} (cos θ) ρ_{d} (cos θ) d cos θ . \end{array}

u^{m} (τ_{L}, + μ) = u_{g}^{m} (μ),

u_{g}^{m} (μ) = δ_{0 m} ∊ (μ) B (T_{g}) + (1 + δ_{0 m}) \int_{0}^{1} μ^{'} ρ_{d}^{m} (μ, - μ^{'}) u^{m} (τ_{L}, - μ^{'}) d μ^{'} + \frac{μ_{0}}{π} I_{0} exp (- τ_{L} / μ_{0}) ρ_{d}^{m} (μ, - μ_{0}) .

u_{1}^{m} (0, - μ_{i}) = u_{\infty}^{m} (- μ_{i}), i = 1, \dots, N,

\begin{matrix} u_{p}^{m} (τ_{p}, μ_{i}) = u_{p + 1}^{m} (τ_{p}, μ_{i}), i = \pm 1, \dots, \pm N; \\ p = 1, \dots, L - 1; \end{matrix}

u_{L}^{m} (τ_{L}, + μ_{i}) = u_{g}^{m} (μ_{i}), i = 1, \dots, N,

u_{g}^{m} (μ_{i}) = δ_{0 m} ∊ (μ_{i}) B (T_{g}) + (1 + δ_{0 m}) \sum_{j = 1}^{N} w_{j} μ_{j} ρ_{d}^{m} (μ_{i}, - μ_{j}) u^{m} (τ_{L}, - μ_{j}) + \frac{μ_{0}}{π} I_{0} exp (- τ_{L} / μ_{0}) ρ_{d}^{m} (μ_{i}, - μ_{0}) .

u_{p} (τ, μ_{i}) = \sum_{j = 1}^{N} [C_{j p} G_{j p} (μ_{i}) exp (- k_{j p} τ) + C_{- j p} G_{- j p} (μ_{i}) exp (+ k_{j p} τ)] + R_{p} (τ, μ_{i}),

R_{p} (τ, μ_{i}) = Z_{0} (μ_{i}) exp (- τ / μ_{0}) + δ_{m 0} [Y_{0} (μ_{i}) + Y_{1} τ] .

\sum_{j = 1}^{N} [C_{j 1} G_{j 1} (- μ_{i}) + C_{- j 1} G_{- j 1} (- μ_{i})] = u_{\infty} (- μ_{i}) - R_{p} (0, - μ_{i}), i = 1, \dots, N

\begin{array}{l} \sum_{j = 1}^{N} {C_{j p} G_{j p} (μ_{i}) exp (- k_{j p} τ_{p}) \\ + C_{- j p} G_{- j p} (μ_{i}) exp (k_{j p} τ_{p}) \\ - [C_{j, p + 1} G_{j, p + 1} (μ_{i}) exp (- k_{j, p + 1} τ_{p}) \\ + C_{- j, p + 1} G_{- j, p + 1} (μ_{i}) exp (k_{j, p + 1} τ_{p})]} \\ = [R_{p + 1} (τ_{p}, μ_{i}) - R_{p} (τ_{p}, μ_{i})], i = \pm 1, \dots \pm N; p = 1, \dots, L - 1 \end{array}

\sum_{j = 1}^{N} [C_{j L} r_{j L} (μ_{i}) exp (- k_{j L} τ_{L}) + C_{- j L} r_{- j L} (μ_{i}) exp (k_{j L} τ_{L})] = γ (τ_{L}, μ_{i}), i = 1, \dots, N,

γ (τ_{L}, μ_{i}) = δ_{0 m} ∊ (μ_{i}) B (T_{g}) - R_{L} (τ_{L}, + μ_{i}) + \frac{μ_{0}}{π} I_{0} exp (- τ_{L} / μ_{0}) ρ_{d} (μ_{i}, μ_{0}) + (1 + δ_{0 m}) \sum_{j = 1}^{N} ρ_{d} (μ_{i}, - μ_{j}) w_{j} μ_{j} R_{L} (τ_{L}, - μ_{j}),

r_{j L} (μ_{i}) = G_{j L} (μ_{i}) - (1 + δ_{0 m}) \times \sum_{n = 1}^{N} ρ_{d} (μ_{i}, - μ_{n}) w_{n} μ_{n} G_{j L} (- μ_{n}) .

C_{+ j p} = {\tilde{C}}_{+ j p} exp (k_{j p} τ_{p - 1}),

C_{- j p} = {\tilde{C}}_{- j p} exp (- k_{j p} τ_{p}) .

u_{p} (τ, μ_{i}) = \sum_{j = 1}^{N} {{\tilde{C}}_{j p} G_{j p} (μ_{i}) exp [- k_{j p} (τ - τ_{p - 1})] + {\tilde{C}}_{- j p} G_{- j p} (μ_{i}) exp [- k_{j p} (τ_{p} - τ)]} .

u (τ, + μ) = u (τ_{L}, + μ) exp [- (τ_{L} - τ) μ] + \int_{τ}^{τ_{L}} S (t, + μ) exp [- (t - τ) / μ] \frac{d t}{μ},

u (τ, - μ) = u (0, - μ) exp (- τ / μ) + \int_{0}^{τ} S (t, - μ) exp [- (τ - t) / μ] \frac{d t}{μ},

\begin{array}{l} u_{p} (τ, + μ) = u (τ_{L}, + μ) exp [- (τ_{L} - τ) / μ] \\ + \sum_{n = p}^{L} {\sum_{j = 1}^{N} [{\tilde{C}}_{- j n} \frac{G_{- j n} (+ μ)}{1 + k_{j n} μ} E_{- j n} (τ, + μ) \\ + \sum {\tilde{C}}_{+ j n} \frac{G_{+ j n} (+ μ)}{1 + k_{j n} μ} E_{+ j n} (τ, + μ)]] \\ + \frac{G_{0 n}}{1 + μ / μ_{0}} E_{0 n} (τ, + μ) \\ + δ_{m 0} [V_{0 n} (μ) F_{0 n} (τ, μ) + b_{1 n} F_{1 n} (τ, μ)]}, \end{array}

E_{- j n} (τ, + μ) = exp [- (k_{j n} Δ τ_{n} - δ τ / μ)] - exp [- (τ_{n} - τ) / μ],

F_{0 n} (τ, + μ) = exp (- δ τ / μ) - exp [- (τ_{n} - τ) / μ],

F_{1 n} (τ, + μ) = (τ_{n - 1} + μ) exp (- δ τ / μ) - (τ_{n} + μ) exp [- (τ_{n} - τ) / μ],

\begin{array}{l} δ τ = τ_{n - 1} - τ for n > p and δ τ = 0 for n = p; \\ Δ τ_{n} = τ_{n} - τ_{n - 1} for n > p and Δ τ_{p} = τ_{p} - τ for n = p, \\ E_{+ j n} (τ, + μ) = exp [- (τ_{n - 1} - τ) / μ] - exp {- [k_{j n} (τ_{n} - τ_{n - 1}) - (τ - τ_{n}) / μ]}, \end{array}

E_{+ j p} (τ, + μ) = exp [- k_{j p} (τ - τ_{p - 1})] - exp {- [k_{j p} (τ_{p} - τ_{p - 1}) + (τ_{p} - τ) / μ]} .

\begin{array}{l} u_{p} (τ, - μ) = u (0, - μ) exp (- τ / μ) \\ + \sum_{n = 1}^{p} {\sum_{j = 1}^{N} [{\tilde{C}}_{- j n} \frac{G_{- j n} (- μ)}{1 + k_{j n} μ} E_{- j n} (τ, - μ) \\ + {\tilde{C}}_{+ j n} \frac{G_{+ j n} (- μ)}{1 - k_{j n} μ} E_{+ j n} (τ, - μ)] \\ + \frac{G_{0 n} (- μ s)}{1 - μ / μ_{0}} E_{0 n} (τ, - μ) \\ + δ_{m 0} [V_{0 n} (μ) F_{0 n} (τ, - μ) + b_{1 n} F_{1 n} (τ, - μ)]}, \end{array}

E_{+ j n} (τ, - μ) = exp [- (k_{j n} Δ τ_{n} + δ τ / μ)] - exp [- (τ - τ_{n - 1}) / μ]

F_{0 n} (τ, - μ) = exp (- δ τ / μ) - exp [- (τ - τ_{n - 1}) / μ]

F_{1 n} (τ, - μ) = (τ_{n} - μ) exp (- δ τ / μ) - (τ_{n - 1} - μ) exp [- (τ - τ_{n - 1}) / μ],

\begin{array}{l} δ τ = τ - τ_{n} for n < p and δ τ = 0 for n = p, \\ Δ τ_{n} = τ_{n} - τ_{n - 1} for n < p and Δ τ = τ_{p - 1} for n = p, \\ E_{- j n} (τ, - μ) = exp [- (τ - τ_{n}) / μ] - exp {- [k_{j n} (τ_{n} - τ_{n - 1}) + (τ - τ_{n - 1}) / μ]}, \end{array}

E_{- j p} (τ, - μ) = exp [- k_{j p} (τ_{p} - τ)] - exp {- [k_{j p} (τ_{p} - τ_{n - 1}) + (τ - τ_{p - 1}) / μ]} .

G_{j} (μ) = \sum_{\begin{array}{l} i = - N \\ i \neq 0 \end{array}}^{N} w_{i} D (μ, μ_{i}) G_{j} (μ_{i}) for j \neq 0;

G_{0} (μ) \equiv Z_{0} (μ) = \sum_{\begin{array}{l} i = - N \\ i \neq 0 \end{array}}^{N} w_{i} D (μ, μ_{i}) Z_{0} (μ_{i}) + X_{0} (μ);

V_{0} (μ) = \sum_{\begin{array}{l} i = - N \\ i \neq 0 \end{array}}^{N} w_{i} D^{0} (μ, μ_{i}) Y_{0} (μ_{i}) + (1 - ω) b_{0};

a (μ) = u^{0} (0, μ); a^{*} (μ) = u^{0 *} (τ_{L}, - μ);

t (μ) = u^{0 *} (0, μ); t^{*} (μ) = u^{0} (τ_{L}, - μ);

Abstract

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