On a Convergence Criterion for Numerical Solvers of the Linear Transport Equation for Neutral Particles

M. Schramm (), C. A. Ladeia () and B. E. J. Bodmann ()
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M. Schramm: Engineering Center of Federal University of Pelotas
C. A. Ladeia: Math Department of Federal University of Rio Grande do Sul
B. E. J. Bodmann: Mechanical Engineering Department of Federal University of Rio Grande do Sul

Chapter Chapter 26 in Integral Methods in Science and Engineering, 2026, pp 399-412 from Springer

Abstract: 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’s coefficients. Then, upon applying the Budan-Fourier theorem to estimate how many roots exist in the two regions ( − ∞ , −1 ) $$(-\infty ,-1)$$ and ( 1 , ∞ ) $$(1, \infty )$$ , 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.

Date: 2026
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DOI: 10.1007/978-3-032-04458-7_26

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