Chord length sampling with memory effects for spatially heterogeneous Markov media: Application to the rod model

In this work we propose a modified Chord Length Sampling (CLS) algorithm, endowed with two layers of “memory effects,” aimed at solving particle transport problems in one-dimensional spatially nonhomogeneous Markov media. CLS algorithms are a family of Monte Carlo methods which account for the stochastic nature of the media by sampling on-the-fly the random interfaces between material phases during the particle propagation. The possibility for the particles to remember the last crossed interfaces increases the accuracy of these models with respect to reference solutions obtained by solving the Boltzmann equation on a large number of realizations of the Markov media. In previous investigations, CLS models with memory have been tested exclusively for spatially uniform stochastic media: in this paper we extend this class of Monte Carlo methods to the case of spatially nonhomogeneous configurations. The effectiveness and the robustness of the modified CLS are probed considering several benchmark problems with varying material cross sections and Markov media densities. The obtained results are a stepping stone towards a generalization to three-dimensional models.

Figure 1

Two realizations of one-dimensional Markov mixtures on the segment $[-L/2,L/2]$, sampled using the sweeping algorithm described in Sec. 2b. Top: Homogeneous Markov mixing of three materials, with transition rates ${\rho }_{0,1}=100/99, {\rho }_{0,2}=10/99, {\rho }_{1,0}=100/11, {\rho }_{1,2}=10/11, {\rho }_{2,0}=100/55$, and ${\rho }_{2,1}$=10/55, and initial conditions $p_0=0.833, p_1=0.085$, and $p_2=0.081$, respectively. Bottom: Homogeneous Markov mixing of two materials, with transition rates ${\rho }_{0,1}=100/99$ and ${\rho }_{1,0}=100/11$, and initial condition $p_0(-L/2)=0.9$. In both cases the rod length is $L=20$.

Figure 2

(a) Coloring probability and (b) forward transition rates as a function of $x$ along the rod. Dots and triangles represent the values obtained from the sampling procedure, while lines represent the expected theoretical values. The binary homogeneous Markov mixture is characterized by rates ${\rho }_{0,1}=10/99, {\rho }_{1,0}=10/11$, with initial condition $p_0(-L/2)=0.9$. Error bars are computed according to Eq. (11).

Figure 3

(a) Coloring probability, (b) transition rates $0\to 1$, (c) transition rates $1\to 0$ as a function of the position $x$ along the rod. Dots and triangles represent the values obtained from the sampling procedure, and lines represent the expected theoretical values. The forward transition rates ${\rho }_{0,1}$ (used for sweeping) follow the ramplike function given in Eq. (14), with parameters $y_{0,1}=10/99, m_{0,1}=0.5, d_{0,1}=5$, and ${\rho }_{1,0}$ is constant along the rod with value $10/11$. Error bars are computed according to Eq. (11).

Figure 4

(a) Coloring probability, (b) transition rates $0\to 1$, (c) transition rates $1\to 0$ as a function of the position $x$ along the rod. Dots and triangles represent the values obtained from the sampling procedure, and lines represent the expected theoretical values. The input transition rates (used for sweeping) are given by Eqs. (15) and (16). Error bars are computed according to Eq. (11).

Figure 5

Coloring probability as a function of the position $x$ along the rod. Dots, triangles, and squares represent the ensemble averages resulting from realizations, and lines represent the expected theoretical values. The ternary heterogeneous Markov mixture is characterized by the mixing statistics given by Eq. (14), whose coefficients are reported in Table 1. Error bars are computed according to Eq. (11).

Figure 6

(a, b) Scalar flux in material 0 (red, upper lines) and 1 (blue, lower lines) as a function of $x$ along the rod, for cases I and II, respectively. Solid line: Reference solutions; dots: algorithm A (memoryless); triangles: algorithm B (one layer of memory effect); crosses: algorithm C (two layers of memory effect).

Figure 7

(a)–(d) Scalar flux in material 0 (red, upper lines) and 1 (blue, lower lines) as a function of $x$ along the rod, for the four simulation cases III, IV, V, and VI, respectively. Solid line: Reference solutions; dots: algorithm A (memoryless); triangles: algorithm B (one layer of memory effect); crosses: algorithm C (two layers of memory effect).

Figure 8

Analysis of the normalized distribution $\mathrm{\Pi }\left(\text{R}\right)$ of the reflection coefficient $R$ and of the normalized distribution $\mathrm{\Pi }\left(\text{T}\right)$ of the transmission coefficients $T$, for the benchmark cases I (top), III (center), and VI (bottom). Distributions have been obtained from the reference simulations over the ensemble of sampled stochastic media. Left column: $\mathrm{\Pi }\left(\text{R}\right)$; right column: $\mathrm{\Pi }\left(\text{T}\right)$. The vertical lines denote the average values: blue solid line reference value, red dashed line CLS A, green dotted line CLS B, violet dash-dotted line CLS C.