Analysis of a Monte Carlo method for nonlinear radiative transfer. (English) Zbl 0628.65128
The authors consider problems of nonlinear radiative transfer and their solutions obtained by the method of J. A. Fleck jun. and J. D. Cummings [ibid. 8, 313-342 (1971; Zbl 0229.65087)]. This method firstly consists in a time discretization; then in every time interval \(\Delta t_ i=t_{i+1}-t_ i\) the nonlinear transport process is approximated by a linear one, which is eventually handled by a standard linear transport Monte Carlo method.
It was recently proved that the solutions of these problems satisfy a maximum principle (that is, under suitable hypotheses, they satisfy, for every \(t>0\), some inequalities, if these are satisfied by the initial values). The authors observe that the approximate solutions obtained by the Fleck-Cummings method may violate this principle if the \(\Delta t_ i\) are not sufficiently small, and give an upper bound for the maximum \(\Delta t_ i\) so that the principle will be surely satisfied.
The solutions the authors consider are “ideal” solutions free from statistical errors and not any actual Monte Carlo solution. The paper contains also numerical applications from which it can be deduced that the theoretical bound is indeed much smaller than the value of max \(\Delta\) \(t_ i\) for which the violation of the principle may occur.
It was recently proved that the solutions of these problems satisfy a maximum principle (that is, under suitable hypotheses, they satisfy, for every \(t>0\), some inequalities, if these are satisfied by the initial values). The authors observe that the approximate solutions obtained by the Fleck-Cummings method may violate this principle if the \(\Delta t_ i\) are not sufficiently small, and give an upper bound for the maximum \(\Delta t_ i\) so that the principle will be surely satisfied.
The solutions the authors consider are “ideal” solutions free from statistical errors and not any actual Monte Carlo solution. The paper contains also numerical applications from which it can be deduced that the theoretical bound is indeed much smaller than the value of max \(\Delta\) \(t_ i\) for which the violation of the principle may occur.
Reviewer: M.Cugiani
MSC:
65Z05 | Applications to the sciences |
65N40 | Method of lines for boundary value problems involving PDEs |
65C05 | Monte Carlo methods |
35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
85A25 | Radiative transfer in astronomy and astrophysics |
Keywords:
nonlinear radiative transfer; time discretization; linear transport Monte Carlo method; maximum principleCitations:
Zbl 0229.65087References:
[1] | Andreev, E. S.; Kozmanov, M. Yu.; Rachilov, E. B., U.S.S.R. Comput. Math. Math. Phys., 23, 104 (1983) |
[3] | Fleck, J. A.; Cummings, J. D., J. Comput. Phys., 8, 313 (1971) · Zbl 0229.65087 |
[4] | Alcouffe, R. E.; Clark, B. A.; Larsen, E. W., (Brackbill, J. U.; Cohen, B., Multiple Time Scales (1985), Academic Press: Academic Press Orlando), 73 · Zbl 0563.00023 |
[5] | Pomraming, G. C., The Equations of Radiation Hydrodynamics (1973), Pergamon: Pergamon Oxford |
[6] | Duderstadt, J. J.; Martin, W. R., Transport Theory (1979), Wiley-Interscience: Wiley-Interscience New York · Zbl 0407.76001 |
[7] | Lewis, E. E.; Miller, W. F., Computational Methods of Neutron Transport (1984), Wiley-Interscience: Wiley-Interscience New York · Zbl 0594.65096 |
[8] | Fleck, J. A.; Canfield, E. H., J. Comput. Phys., 54, 508 (1984) · Zbl 0558.65096 |
[9] | Case, K. M.; Zweifel, P. F., Linear Transport Theory, ((1967), Addison-Wesley: Addison-Wesley Reading, MA), 282 · Zbl 0132.44902 |
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