Article Contents
Article Contents

# Numerical solutions for multidimensional fragmentation problems using finite volume methods

• We introduce a finite volume scheme for approximating a general multidimensional fragmentation problem. The scheme estimates several physically significant moment functions with good accuracy, and is robust with respect to use of different nonuniform daughter distribution functions. Moreover, it possess simple mathematical formulation for defining in higher dimensions. The efficiency of the scheme is validated over several test problems.

Mathematics Subject Classification: Primary: 45K05, 65R20; Secondary: 45L05.

 Citation:

• Figure 1.  Exact and numerical values of the normalized moments

Figure 2.  Exact and numerical values of the normalized moments with size independent selection function

Figure 3.  Exact and numerical values of the normalized moments with size dependent selection function

Figure 4.  Exact and numerical values of the normalized moments with size independent selection function

Figure 5.  Exact and numerical values of the normalized moments with size dependent selection function

Figure 6.  Exact and numerical values of the normalized moments with the kernels having three particle properties

Table 1.  Summary of the selected test problems in two dimensions

 Test case $S(x_1,x_2)$ $b(x_1,x_2|y_1,y_2)$ Exact moments $1$ $1$ $\frac{2}{y_1y_2}$ $\mathcal{M}_{k,l}(t) = \exp\left[\left(\frac{2}{(k+1)(l+1)} - 1\right)t\right]$ $2$ $1$ $2\delta\left(x_1-\frac{y_1}{2}\right)\delta\left(x_2-\frac{y_2}{2}\right)$ $\mathcal{M}_{k,l}(t) = \exp\left[\left(2^{1-k-l} - 1\right)t\right]$ $3$ $x_1+x_2$ $\frac{2}{y_1y_2}$ $\mathcal{M}_{1,0}(t) = \mathcal{M}_{0,1}(t) = 1$, $\mathcal{M}_{0,0} (t) = 1 + 2t$ $4$ $x_1+x_2$ $2\delta\left(x_1-\frac{y_1}{2}\right)\delta\left(x_2-\frac{y_2}{2}\right)$ $\mathcal{M}_{1,0}(t) = \mathcal{M}_{0,1}(t) = 1$, $\mathcal{M}_{0,0} (t) = 1 + 2t$ $5$ $1$ $\frac{4}{y_1y_2}$ $\mathcal{M}_{k,l}(t) = \exp\left[\left(\frac{4}{(k+1)(l+1)} - 1\right)t\right]$ $6$ $1$ $\frac{y_1\delta\left(x_1-y_1\right) + y_2\delta\left(x_2-y_2\right)}{y_1y_2}$ $\mathcal{M}_{1,1} (t) = 1$, $\mathcal{M}_{0,0} (t) = \exp(t)$, $\mathcal{M}_{1,0}(t) + \mathcal{M}_{0,1}(t) =\exp(t/2)$ $7$ $x_1+x_2$ $\frac{4}{y_1y_2}$ $\mathcal{M}_{1,1}(t) = 1$, $\mathcal{M}_{0,0}(t) = 1 + 3t$ $8$ $x_1+x_2$ $\frac{y_1\delta\left(x_1-y_1\right) + y_2\delta\left(x_2-y_2\right)}{y_1y_2}$ $\mathcal{M}_{1,1}(t) = 1$, $\mathcal{M}_{0,0}(t) = 1 + t$

Table 2.  Summary of the selected test problems in three dimensions

 Test case $S(x_1,x_2, x_3)$ $b(x_1,x_2,x_3|y_1,y_2,y_3)$ Exact moments $9$ $x_1+x_2+x_3$ $2\delta\left(x_1-\frac{y_1}{2}\right)\delta\left(x_2-\frac{y_2}{2}\right) \delta\left(x_3-\frac{y_3}{2}\right)$ $\mathcal{M}_{1,0,0}(t) = \mathcal{M}_{1,0,1}(t) = 1$, $\mathcal{M}_{0,0,1}(t) = 1$, $\mathcal{M}_{0,0,0}(t) = 1+ 3t$ $10$ $x_1x_2x_3$ $\frac{8}{y_1y_2y_3}$ $\mathcal{M}_{1,1,1}(t) = 1$, $\mathcal{M}_{0,0,0}(t) = 1+ 7t$

Table 3.  Relative error for the weighted moments at different times for the test case $1$

 Scheme$-1a$ Scheme$-2a$ $t$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ $\mu_{1,1}(t)$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ $\mu_{1,1}(t)$ 1 0.40989 3.3307E-16 0.19613 4.5897E-06 1.7764E-15 0.16397 2 0.65177 2.2204E-16 0.43072 2.5283E-06 1.7764E-15 0.30105 3 0.79452 2.2204E-16 0.51132 2.8065E-05 1.3323E-15 0.41566 4 0.87877 4.4409E-16 0.62470 4.6254E-05 1.3323E-15 0.51147 5 0.84305 2.2204E-16 0.76290 1.4383E-05 4.4409E-16 0.59157

Table 4.  Relative error for the weighted moments at different times for the test case $2$

 Scheme$-1a$ Scheme$-2a$ $t$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ $\mu_{1,1}(t)$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ $\mu_{1,1}(t)$ 1 0.35822 4.3652E-09 0.51745 5.8726E-09 4.3652E-09 0.44845 2 0.84477 4.3652E-08 0.54865 6.5772E-08 4.3652E-09 0.41034 3 1.5056 4.3652E-08 0.57071 8.5278E-08 4.3652E-09 0.36958 4 2.1524 4.3652E-08 0.59846 6.8704E-08 4.3652E-09 0.33718 5 3.2816 4.3652E-08 0.62442 6.7941E-08 4.3652E-09 0.29138

Table 5.  Relative error for higher order weighted moments using different computational grids for the test case $1$ at $t = 5$

 Scheme$-1a$ Scheme$-2a$ (Grids) (Grids) Moments $15\times15$ $20\times20$ $25\times25$ $15\times15$ $20\times20$ $25\times25$ $\mu_{2,0}(t)$ 0.12213 0.12209 3.7968E-02 0.20312 0.26686 0.28581 $\mu_{0,2}(t)$ 0.12213 0.12209 3.7968E-02 0.20312 0.26686 0.28581 $\mu_{3,0}(t)$ 0.39737 0.11985 4.3010E-02 0.21663 0.27822 8.4753E-02 $\mu_{2,1}(t)$ 0.53920 0.53003 0.48809 0.87975 0.80544 0.73234 $\mu_{1,2}(t)$ 0.53920 0.53003 0.48809 0.87975 0.80544 0.73234 $\mu_{3,0}(t)$ 0.39737 0.11985 4.3010E-02 0.21663 0.27822 8.4753E-02

Table 6.  Relative error for higher order weighted moments using different computational grids for the test case $2$ at $t = 5$

 Scheme$-1a$ Scheme$-2a$ (Grids) (Grids) Moments $15\times15$ $20\times20$ $25\times25$ $15\times15$ $20\times20$ $25\times25$ $\mu_{2,0}(t)$ 0.46289 0.43660 0.43344 0.43994 0.29138 5.1652E-02 $\mu_{0,2}(t)$ 0.46289 0.43660 0.43344 0.43994 0.29138 5.1652E-02 $\mu_{3,0}(t)$ 0.57837 0.53984 0.40542 0.58212 0.31483 0.19485 $\mu_{2,1}(t)$ 0.57837 0.53984 0.40542 0.58212 0.31483 0.19485 $\mu_{1,2}(t)$ 0.57837 0.53984 0.40542 0.58212 0.31483 0.19485 $\mu_{3,0}(t)$ 0.57837 0.53984 0.40542 0.58212 0.31483 0.19485

Table 7.  Relative error for the weighted moments at different times for the test case $3$

 Scheme$-1a$ Scheme$-2a$ $t$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ 1 0.29315 2.2204E-16 7.5670E-08 2.2204E-16 2 0.37684 2.2204E-16 1.2102E-08 2.2204E-16 3 0.41648 4.4409E-16 7.1239E-07 2.2204E-16 4 0.43960 1.3323E-15 2.6133E-07 4.4409E-16 5 0.45475 1.1102E-15 7.1965E-06 2.2204E-16

Table 8.  Relative error for the weighted moments at different times for the test case $4$

 Scheme$-1a$ Scheme$-2a$ $t$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ 1 0.15457 4.3652E-15 1.5366E-16 4.3652E-15 2 0.21110 4.3652E-15 1.2179E-16 4.3652E-15 3 0.23833 4.3652E-15 1.7218E-16 4.3652E-15 4 0.24726 4.3652E-15 1.3315E-16 4.3652E-15 5 0.25532 4.3652E-15 2.1709E-16 4.3652E-15

Table 9.  CPU usage time (in seconds) taken to solve test cases 3 and 4

 Method Test case 3 Test case 4 Scheme−1a 1 4 Scheme−2a 1 7

Table 10.  Relative error for the weighted moments at different times for the test case $5$

 Scheme$-1b$ Scheme$-2b$ $t$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ $\mu_{1,1}(t)$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ $\mu_{1,1}(t)$ 1 0.83984 0.40989 3.3307E-16 2.7510E-06 0.23775 6.6613E-16 2 0.97435 0.65177 4.4409E-16 1.3740E-06 0.53049 4.4409E-16 3 0.99590 0.79452 1.2212E-15 3.1276E-05 0.87720 1.7764E-15 4 0.99935 0.87877 1.8874E-15 4.7495E-05 0.92423 1.5543E-15 5 0.99990 0.92852 1.9984E-15 1.4680E-05 0.95464 1.7764E-15

Table 11.  Relative error for the weighted moments at different times for the test case $6$

 Scheme$-1b$ Scheme$-2b$ $t$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ $\mu_{1,1}(t)$ $\mu_{0,0}(t)$ $\mu_{1,0}(t)+\mu_{0,1}(t)$ $\mu_{1,1}(t)$ 1 0.24477 0.11131 4.8411E-15 9.7725E-07 1.4316E-03 4.8411E-15 2 0.42963 0.21084 4.8411E-15 1.0209E-06 3.1159E-02 4.8411E-15 3 0.56924 0.35725 4.8411E-15 1.3395E-06 5.9667E-02 4.8411E-15 4 0.65103 0.51075 4.8411E-15 1.6825E-06 9.3380E-01 4.8411E-15 5 0.73647 0.79839 4.8411E-15 1.7411E-06 1.6495E-01 4.8411E-15

Table 12.  Relative error for higher order weighted moments using different computational grids for the test case $5$ at $t = 5$

 Scheme$-1a$ Scheme$-2a$ (Grids) (Grids) Moments $15\times15$ $20\times20$ $25\times25$ $15\times15$ $20\times20$ $25\times25$ $\mu_{2,0}(t)$ 0.66871 0.54513 0.31947 4.2865 1.7192 0.81177 $\mu_{0,2}(t)$ 0.66871 0.54513 0.31947 4.2865 1.7192 0.81177 $\mu_{3,0}(t)$ 3.1432 2.0981 0.40542 10.784 3.8650 1.7161 $\mu_{2,1}(t)$ 0.43645 0.47180 0.47048 0.59341 0.58034 0.54996 $\mu_{1,2}(t)$ 0.43645 0.47180 0.47048 0.59341 0.58034 0.54996 $\mu_{3,0}(t)$ 3.1432 2.0981 1.1880 10.784 3.8650 1.7161

Table 13.  Relative error for the weighted moments at different times for the test case $7$

 Scheme$-1b$ Scheme$-2b$ $t$ $\mu_{0,0}(t)$ $\mu_{1,1}(t)$ $\mu_{0,0}(t)$ $\mu_{1,1}(t)$ 1 0.32971 2.2204E-16 2.0426E-16 2.2204E-16 2 0.42818 2.2204E-16 2.6527E-16 4.4409E-16 3 0.47552 2.2204E-16 1.9640E-16 2.2204E-16 4 0.50335 2.2204E-16 1.5592E-16 2.2204E-16 5 0.52166 4.4409E-16 2.5855E-16 1.1102E-16

Table 14.  Relative error for the weighted moments at different times for the test case $8$

 Scheme$-1b$ Scheme$-2b$ $t$ $\mu_{0,0}(t)$ $\mu_{1,1}(t)$ $\mu_{0,0}(t)$ $\mu_{1,1}(t)$ 1 9.3859E-02 4.8411E-16 1.4390E-16 4.8411E-16 2 0.13885 4.8411E-16 2.1288E-16 4.8411E-16 3 0.15973 4.8411E-16 1.7811E-16 4.8411E-16 4 0.17886 4.8411E-16 4.3876E-16 4.8411E-16 5 0.19219 4.8411E-16 1.2407E-16 4.8411E-16

Table 15.  Relative error for the weighted moments at different times for the test case $9$

 Scheme$-1a$ Scheme$-2a$ $t$ $\mu_{0,0,0}(t)$ $\mu_{1,0,0}(t)+\mu_{0,1,0}(t)+\mu_{0,0,1}(t)$ $\mu_{0,0,0}(t)$ $\mu_{1,0,0}(t)+\mu_{0,1,0}(t)+\mu_{0,0,1}(t)$ 1 0.55768 1.0322E-16 1.4147E-16 1.0322E-16 2 0.66334 1.0322E-16 1.6827E-16 1.0322E-16 3 0.69948 1.0322E-15 1.1974E-16 1.0322E-16 4 0.72767 1.0322E-16 1.9689E-16 1.0322E-16 5 0.74506 1.0322E-16 4.7747E-16 1.0322E-16

Table 16.  Relative error for the weighted moments at different times for the test case $10$

 Scheme$-1b$ Scheme$-2b$ $t$ $\mu_{0,0,0}(t)$ $\mu_{1,1,1}(t)$ $\mu_{0,0,0}(t)$ $\mu_{1,1,1}(t)$ 1 0.75665 1.1102E-16 4.2454E-16 2.2204E-16 2 0.80306 1.1102E-16 3.3864E-16 2.2204E-16 3 0.81597 1.1102E-16 1.6708E-16 2.2204E-16 4 0.82467 2.2204E-16 9.9267E-16 2.2204E-16 5 0.82915 1.1102E-16 3.9475E-16 4.4409E-16

Table 17.  Computational time taken in seconds by the schemes

 Test case $9$ Test case $10$ Scheme$-1a$ Scheme$-2a$ Scheme$-1b$ Scheme$-2b$ 58 86 13 26
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Tables(17)