@conference {10.1007/978-3-032-04458-7_26, title = {On a Convergence Criterion for Numerical Solvers of the Linear Transport Equation for Neutral Particles}, booktitle = {Integral Methods in Science and Engineering}, year = {2026}, note = {n/a}, pages = {399{\textendash}412}, publisher = {Springer Nature Switzerland}, organization = {Springer Nature Switzerland}, address = {Cham}, abstract = {

The transport equation for a scattering medium is an integro-differential equation, which is commonly solved by the use of numerical schemes. A related issue, which is rarely addressed in these kind of approaches is the pertinent question of convergence. Many mesh structured schemes, like finite differences and finite volumes typically make use of a line-by-line iteration. In this direction, the present work focuses on the question whether it is in principal possible to determine, if a numerical scheme is convergent for a specific computational resource, where the iteration scheme is to be executed. To this end an iterative scheme using the finite difference and discrete ordinates method was implemented, where the solution of this scheme is given by a linear algebra system, which is also the case for other numerical approaches. Thus generally speaking, the scheme converges if and only if the spectral radius of the coefficient matrix is less than one. Since the spectral radius is the norm of the eigenvalues, i.e. the roots of the characteristic polynomial, we used proper collocation points and the inverse discrete Fourier transform matrix to obtain the polynomial{\textquoteright}s coefficients. Then, upon applying the Budan-Fourier theorem to estimate how many roots exist in the two regions (-$\infty$,-1){\$}{\$}(-{\backslash}infty ,-1){\$}{\$}and (1,$\infty$){\$}{\$}(1, {\backslash}infty ){\$}{\$}, respectively, one can infer on the spectral radius and thus the convergence of the numerical scheme. In summary, if the two root test results in: (a) (0, 0), then the scheme is convergent; (b) (odd, any), then the scheme is divergent; (c) (even, even \> 0), then the test is inconclusive. This test is performed without explicitly allocating the coefficient matrix. As results of this analysis we show graphs for some cases with regions of convergence for combinations of two parameters. Although this result is conclusive with respect to convergence, one observes that computational complexity leads to an exponential growth, while most of the iteration schemes are of polynomial growth, so that such a test so far takes too much time even for coarse meshes. Hence, we discuss some perspectives to circumvent this shortcoming in order to open pathways for the use of the presented convergence criterion in applications.

}, isbn = {978-3-032-04458-7}, author = {M. Schramm and C. A. Ladeia and B. E. J. Bodmann}, editor = {Constanda, Christian and Bardo E. J. Bodmann and Harris, Paul J.} } @conference {10.1007/978-3-032-04458-7_14, title = {On a Parametrization for the Layer Thickness Depending on the Transmissivity in a Radiative Transfer Problem}, booktitle = {Integral Methods in Science and Engineering}, year = {2026}, note = {n/a}, pages = {199{\textendash}210}, publisher = {Springer Nature Switzerland}, organization = {Springer Nature Switzerland}, address = {Cham}, abstract = {

We consider the radiative transfer equation in Cartesian geometry in a one-dimensional heterogeneous participant medium with two layers. The numerical solutions to the linear problem is discussed and obtained by a combination of the discrete ordinates and the finite difference methods. The objective of this work is to show a simple parametric formula for the thickness of one of the layers for a specified transmissivity. Further, we present error estimates for the considered steady-state transport equations with azimuthal symmetry and isotropic scattering.

}, isbn = {978-3-032-04458-7}, author = {C. A. Ladeia and Guimar{\~a}es, A. B. and M. Schramm and B. E. J. Bodmann}, editor = {Constanda, Christian and Bardo E. J. Bodmann and Harris, Paul J.} } @article {Schramm_Ladeia_Fernandes_2024, title = {Solution of Linear Radiative Transfer Equation in Hollow Sphere by Diamond Difference Discrete Ordinates and Decomposition Methods}, journal = {Semina: Ci{\^e}ncias Exatas e Tecnol{\'o}gicas}, volume = {45}, year = {2024}, note = {n/a}, month = {Dec.}, pages = {e51961}, abstract = {n/a}, doi = {10.5433/1679-0375.2024.v45.51961}, url = {https://ojs.uel.br/revistas/uel/index.php/semexatas/article/view/51961}, author = {Schramm, Marcelo and Ladeia, Cibele and Fernandes, Julio Cesar} } @conference {10.1007/978-3-031-34099-4_15, title = {The Influence of the Refractive Index and Absorption Coefficients in the Solution of the Radiative Conductive Transfer Equation in Cartesian Geometry}, booktitle = {Integral Methods in Science and Engineering}, year = {2023}, note = {n/a}, pages = {179{\textendash}189}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {n/a}, isbn = {978-3-031-34099-4}, author = {C. A. Ladeia and M. Schramm and J. C. L. Fernandes and Zanetti, H. R. and Albuquerque, A. D.}, editor = {Constanda, Christian and Bardo E. J. Bodmann and Harris, Paul J.} } @conference {10.1007/978-3-031-07171-3_12, title = {A Simple Numerical Scheme to Obtain Reflectivity and Transmissivity of an Isotropically Scattering Slab}, booktitle = {Integral Methods in Science and Engineering}, year = {2022}, note = {n/a}, pages = {169{\textendash}178}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {

In this chapter, we consider the linear radiative transfer model for a passive medium with no intrinsic thermal contributions. The domain is a one-dimensional slab for which reflectivity and transmissivity are determined from the total in- and outgoing radiative fluxes across the boundaries. Further, we assume isotropic scattering in the medium and no emissivity and reflectivity on the boundaries but thermal radiation from the latter and an isotropic incoming radiation from an external source. Our developments are based on the discrete ordinate method in the angular variable and a modified version of the finite volume method in the spatial variable. The solution is obtained by an iterative method, and we present a necessary condition for its convergence based on norm operations.

}, isbn = {978-3-031-07171-3}, author = {C. A. Ladeia and Zanetti, H. R. and Gisch, D. L. and M. Schramm and J. C. L. Fernandes}, editor = {Constanda, Christian and Bardo E. J. Bodmann and Harris, Paul J.} } @article {Ladeia2020, title = {Radiative{\textendash}conductive transfer equation in spherical geometry: arithmetic stability for decomposition using the condition number criterion}, journal = {Journal of Engineering Mathematics}, volume = {123}, year = {2020}, note = {

n/a

}, pages = {149-163}, abstract = {

n/a

}, doi = {10.1007/s10665-020-10059-2}, author = {C. A. Ladeia and Bardo E. J. Bodmann and Marco T. Vilhena} } @article {Ladeia2020, title = {A simple numerical scheme to linear radiative transfer in hollow and solid spheres}, journal = {SEMINA. CI{\^E}NCIAS EXATAS E TECNOL{\'O}GICAS}, year = {2020}, note = {n/a}, abstract = {n/a}, doi = {10.5433/1679-0375.2020v41n1p21}, author = {C. A. Ladeia and M. Schramm and J. C. L. Fernandes} } @article {Ladeia2019, title = {On the Integro-Differential Radiative Conductive Transfer Equation: A Modified Decomposition Method}, journal = {Integral Methods in Science and Engineering}, year = {2019}, note = {n/a}, abstract = {n/a}, doi = {10.1007/978-3-030-16077-7_16}, author = {C. A. Ladeia and B. E. J. Bodmann and M. T. Vilhena} } @article {Ladeia2018118, title = {The Radiative Conductive Transfer Equation in Cylinder Geometry: Rocket Launch Exhaust Phenomena for the Alc{\^a}ntara Launch Center}, journal = {American Journal of Environmental Engineering}, volume = {8}, year = {2018}, note = {n/a}, pages = {118-127}, abstract = {n/a}, doi = {10.5923/j.ajee.20180804.06}, author = {C. A. Ladeia and B. E. J. Bodmann and M. T. Vilhena} } @article {Ladeia2018338, title = {The radiative conductive transfer equation in cylinder geometry: Semi-analytical solution and a point analysis of convergence}, journal = {Journal of Quantitative Spectroscopy and Radiative Transfer}, volume = {217}, year = {2018}, note = {n/a}, pages = {338{\textendash}352}, abstract = {n/a}, doi = {10.1016/j.jqsrt.2018.06.012}, author = {C. A. Ladeia and B. E. J. Bodmann and M. T. Vilhena} } @article {Ladeia2017, title = {On the Radiative Conductive Transfer Equation: A Heuristic Convergence Criterion by Stability Analysis}, journal = {Integral Methods in Science and Engineering}, volume = {1}, year = {2017}, note = {n/a}, abstract = {n/a}, doi = {10.1007/978-3-319-59384-5_15}, author = {C. A. Ladeia and J. C. L. Fernandes and B. E. J. Bodmann and M. T. Vilhena} }