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Radiative–conductive transfer equation in spherical geometry: arithmetic stability for decomposition using the condition number criterion

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Abstract

The radiative–conductive transfer equation in the \(S_N\) approximation for spherical geometry is solved using a modified decomposition method. The focus of this work is to show how to distribute the source terms in the recursive equation system in order to guarantee arithmetic stability and thus numerical convergence of the obtained solution, guided by a condition number criterion. Some examples are compared with results from literature and parameter combinations are analyzed, for which the condition number analysis indicates convergence of the solutions obtained by the recursive scheme.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil

    Cibele A. Ladeia, Bardo E. J. Bodmann & Marco T. Vilhena

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  1. Cibele A. Ladeia
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  2. Bardo E. J. Bodmann
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  3. Marco T. Vilhena
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Correspondence to Cibele A. Ladeia.

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Appendix

Appendix

To obtain the maximum norm of \(\mathbf{J}\), we use the \(S_8\) approximation. Note that the discrete directions and the weights satisfy \(|\mu _i|=\mu _{N+1-i}\) , \(\varpi _i = \varpi _{N+1-i}\), for \(i \in [1,4]\), respectively. Furthermore, we consider isotropic scattering (\(\mathfrak {L} = 0\)). The weights obey the following relations \(\varpi _1 < \varpi _j\) with \(j \in [1,3]\):

Then, the maximum norm of matrix \(\mathbf{J}\), with dominant block \(\mathcal {J}\), is

$$\begin{aligned} \kappa = \Vert J\Vert _{\infty } = 4 \mathcal {J} (\omega -1 )\left( \sum _{\ell =0}^{\mathcal {J}} \varTheta _\ell (r)\right) ^3 + 4(\mathcal {J}-1)(\omega -1 )\left( \sum _{\ell =0}^{\mathcal {J}-1} \varTheta _\ell (r)\right) ^3 + 1- \frac{\omega }{2}\varpi _{1} + \frac{\omega }{2}\sum _{p=2}^{N}\varpi _{p} ; \end{aligned}$$
(30)

otherwise, the contribution of Z in the recursive scheme should be analyzed together with the diagonal blocks behaviors. In the following, we illustrate the Jacobian matrix of order \(N^{2}\mathcal {J}^{2} \times N^{2}\mathcal {J}^{2}\).

Some of the blocks of matrices are null matrices, \(\mathbf {0} = \mathbf{J}_{01}= \mathbf{J}_{0(Z-1)}=\mathbf{J}_{1(Z-1)}=\mathbf{J}_{0Z}=\mathbf{J}_{1Z}=\mathbf{J}_{(Z-1)Z}=\mathbf{J}_{0(Z+1)}= \mathbf{J}_{1(Z+1)} = \mathbf{J}_{(Z-1)(Z+1)}= \mathbf{J}_{Z(Z+1)}= \mathbf{J}_{0(\mathcal {J}-1)}= \mathbf{J}_{1(\mathcal {J}-1)}=J_{(Z-1)(\mathcal {J}-1)}=\mathbf{J}_{Z(\mathcal {J}-1)}=\mathbf{J}_{(Z+1)(\mathcal {J}-1)}= \mathbf{J}_{0\mathcal {J}}=\mathbf{J}_{1\mathcal {J}}=\mathbf{J}_{(Z-1)\mathcal {J}}=\mathbf{J}_{Z\mathcal {J}}=\mathbf{J}_{(Z+1)\mathcal {J}}=\mathbf{J}_{(\mathcal {J}-1)\mathcal {J}}\), so that the Jacobian matrix for this problem is a lower triangular matrix.

$$\begin{aligned} \mathbf{J} = \left[ \begin{array}{ccccccccc } \mathbf{J}_{00} &{} \quad \mathbf {0} &{} \quad \cdots &{} \quad \mathbf {0} &{} \quad \mathbf {0} &{} \quad \mathbf {0} &{} \quad \cdots &{} \quad \mathbf {0} &{} \quad \mathbf {0} \\ \mathbf{J}_{10} &{} \quad \mathbf{J}_{11}&{} \quad \cdots &{} \quad \mathbf {0} &{} \quad \mathbf {0} &{} \quad \mathbf {0} &{} \quad \cdots &{} \quad \mathbf {0} &{} \quad \mathbf {0} \\ \vdots &{} \quad \vdots &{} \quad \ddots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \ddots &{} \quad \vdots &{} \quad \vdots \\ \mathbf{J}_{(Z-1)0} &{} \quad \mathbf{J}_{(Z-1)1} &{} \quad \cdots &{} \quad \mathbf{J}_{(Z-1)(Z-1)} &{} \quad \mathbf {0} &{} \quad \mathbf {0} &{} \quad \cdots &{} \quad \mathbf {0} &{} \quad \mathbf {0} \\ \mathbf{J}_{Z0} &{} \quad \mathbf{J}_{Z1} &{} \quad \cdots &{} \quad \mathbf{J}_{Z(Z-1)} &{} \quad \mathbf{J}_{ZZ} &{} \quad \mathbf {0} &{} \quad \cdots &{} \quad \mathbf {0}&{} \quad \mathbf {0} \\ \mathbf{J}_{(Z+1)0} &{} \quad \mathbf{J}_{(Z+1)1} &{} \quad \cdots &{} \quad \mathbf{J}_{(Z+1)(Z-1)} &{} \quad \mathbf{J}_{(Z+1)Z} &{} \quad \mathbf{J}_{(Z+1)(Z+1)} &{} \quad \cdots &{} \quad \mathbf {0}&{} \quad \mathbf {0} \\ \vdots &{} \quad \vdots &{} \quad \ddots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \ddots &{} \quad \vdots &{} \quad \vdots \\ \mathbf{J}_{(\mathcal {J}-1)0} &{} \quad \mathbf{J}_{(\mathcal {J}-1)1} &{} \quad \cdots &{} \quad \mathbf{J}_{(\mathcal {J}-1)(Z-1)} &{} \quad \mathbf{J}_{(\mathcal {J}-1)Z} &{} \quad \mathbf{J}_{(\mathcal {J}-1)(Z+1)} &{} \quad \cdots &{} \quad \mathbf{J}_{(\mathcal {J}-1)(\mathcal {J}-1)}&{} \quad \mathbf {0} \\ \mathbf{J}_{\mathcal {J}0} &{} \quad \mathbf{J}_{\mathcal {J}1} &{} \quad \cdots &{} \quad \mathbf{J}_{\mathcal {J}(Z-1)} &{} \quad \mathbf{J}_{\mathcal {J}Z} &{} \quad \mathbf{J}_{\mathcal {J}(Z+1)} &{} \quad \cdots &{} \quad \mathbf{J}_{\mathcal {J}(\mathcal {J}-1)} &{} \quad \mathbf{J}_{\mathcal {J}\mathcal {J}} \end{array} \right] \end{aligned}$$
(31)

The diagonal block matrices are generically given by

$$\begin{aligned} \mathbf{J}_{kk} = \left[ \begin{array}{ccccc} 1 &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0\\ (\omega - 1)4(\varTheta _0 + \varTheta _1+ \ldots +\varTheta _k )^3 &{} \quad 1-\frac{\omega }{2}\sum _{l=0}^{\mathfrak {L}}\beta _{l}\mathcal {P}_{l}(\mu _{1})\varpi _{1}\mathcal {P}_{l}(\mu _{1}) &{} \quad -\frac{\omega }{2}\sum _{l=0}^{\mathfrak {L}}\beta _{l}\mathcal {P}_{l}(\mu _{2})\varpi _{2}\mathcal {P}_{l}(\mu _{2}) &{} \quad \cdots &{} \quad -\frac{\omega }{2}\sum _{l=0}^{\mathfrak {L}}\beta _{l}\mathcal {P}_{l}(\mu _{N})\varpi _{N}\mathcal {P}_{l}(\mu _{N})\\ (\omega - 1)4(\varTheta _0 + \varTheta _1+ \ldots +\varTheta _k )^3 &{} \quad -\frac{\omega }{2}\sum _{l=0}^{\mathfrak {L}}\beta _{l}\mathcal {P}_{l}(\mu _{1})\varpi _{1}\mathcal {P}_{l}(\mu _{1}) &{} \quad 1-\frac{\omega }{2}\sum _{l=0}^{\mathfrak {L}}\beta _{l}\mathcal {P}_{l}(\mu _{2})\varpi _{2}\mathcal {P}_{l}(\mu _{2}) &{} \quad \cdots &{} \quad -\frac{\omega }{2}\sum _{l=0}^{\mathfrak {L}}\beta _{l}\mathcal {P}_{l}(\mu _{N})\varpi _{N}\mathcal {P}_{l}(\mu _{N})\\ \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \ddots &{} \quad \vdots \\ (\omega - 1)4(\varTheta _0 + \varTheta _1+ \ldots +\varTheta _k )^3 &{} \quad -\frac{\omega }{2}\sum _{l=0}^{\mathfrak {L}}\beta _{l}\mathcal {P}_{l}(\mu _{1})\varpi _{1}\mathcal {P}_{l}(\mu _{1}) &{} \quad -\frac{\omega }{2}\sum _{l=0}^{\mathfrak {L}}\beta _{l}\mathcal {P}_{l}(\mu _{2})\varpi _{2}\mathcal {P}_{2}(\mu _{2}) &{} \quad \cdots &{} \quad 1-\frac{\omega }{2}\sum _{l=0}^{\mathfrak {L}}\beta _{l}\mathcal {P}_{l}(\mu _{N})\varpi _{N}\mathcal {P}_{l}(\mu _{N})\\ \end{array} \right] \end{aligned}$$

Another set of block matrices are equal, \(\mathbf{J}_{(\mathcal {J}-1)0}= \mathbf{J}_{(\mathcal {J}-1)1}=\cdots = \mathbf{J}_{(\mathcal {J}-1)(Z-1)}= \mathbf{J}_{(\mathcal {J}-1)Z}=\mathbf{J}_{(\mathcal {J}-1)(Z+1)}\), and \(\mathbf{J}_{\mathcal {J}0} = \mathbf{J}_{\mathcal {J}1}= \ldots = \mathbf{J}_{\mathcal {J}(Z-1)}= \mathbf{J}_{\mathcal {J}Z}= \mathbf{J}_{\mathcal {J}(Z+1)}\), with

$$\begin{aligned} \mathbf{J}_{k0} = \left[ \begin{array}{ccccc} 0 &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0 \\ (\omega - 1)4[(\varTheta _0 + \varTheta _1+ \ldots +\varTheta _k )^3 - (\varTheta _0 + \varTheta _1+ \ldots +\varTheta _{k-1} )^3] &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0\\ (\omega - 1)4[(\varTheta _0 + \varTheta _1+ \ldots +\varTheta _k )^3 - (\varTheta _0 + \varTheta _1+ \ldots +\varTheta _{k-1} )^3] &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0\\ \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \ddots &{} \quad \vdots \\ (\omega - 1)4[(\varTheta _0 + \varTheta _1+ \ldots +\varTheta _k )^3 - (\varTheta _0 + \varTheta _1+ \ldots +\varTheta _{k-1} )^3] &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0\\ \end{array} \right] . \end{aligned}$$

For \(j \in [1, k-1]\)

$$\begin{aligned} \mathbf{J}_{(k+1)j} = \left[ \begin{array}{ccccc} 0 &{} \quad \frac{\alpha }{4\pi }\mu _1\varpi _{1}(R-r) &{} \quad \frac{\alpha }{4\pi }\mu _2\varpi _{2}(R-r) &{} \quad \cdots &{} \quad \frac{\alpha }{4\pi }\mu _N\varpi _{N}(R-r) \\ (\omega - 1)4[(\varTheta _0 + \varTheta _1+ \ldots +\varTheta _k )^3 - (\varTheta _0 + \varTheta _1+ \ldots +\varTheta _{k-1} )^3] &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0\\ (\omega - 1)4[(\varTheta _0 + \varTheta _1+ \ldots +\varTheta _k )^3 - (\varTheta _0 + \varTheta _1+ \ldots +\varTheta _{k-1} )^3] &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0\\ \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \ddots &{} \quad \vdots \\ (\omega - 1)4[(\varTheta _0 + \varTheta _1+ \ldots +\varTheta _k )^3 - (\varTheta _0 + \varTheta _1+ \ldots +\varTheta _{k-1} )^3] &{} \quad 0 &{} \quad 0 &{} \quad \cdots &{} \quad 0\\ \end{array} \right] \end{aligned}$$

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Ladeia, C.A., Bodmann, B.E.J. & Vilhena, M.T. Radiative–conductive transfer equation in spherical geometry: arithmetic stability for decomposition using the condition number criterion. J Eng Math 123, 149–163 (2020). https://doi.org/10.1007/s10665-020-10059-2

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