Numerical solutions for multidimensional fragmentation problems using finite volume methods

Jitraj Saha; Nilima Das; Jitendra Kumar; Andreas Bück

doi:10.3934/krm.2019004

Article Contents

2019, Volume 12, Issue 1: 79-103. Doi: 10.3934/krm.2019004

This issue Previous Article A stochastic algorithm without time discretization error for the Wigner equation Next Article On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system

Numerical solutions for multidimensional fragmentation problems using finite volume methods

1.
Department of Mathematics, National Institute of Technology Tiruchirappalli, Tiruchirappalli-620 015, Tamil Nadu, India
2.
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721 302, West Bengal, India
3.
Institute of Particle Technology (LFG), Friedrich-Alexander University Erlangen-Nürnberg, D-91058 Erlangen, Germany

* Corresponding author: Jitraj Saha: jitraj@nitt.edu
* Corresponding author: Jitraj Saha: jitraj@nitt.edu

Received: August 31, 2017

Revised: March 31, 2018

Published: July 2018

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Abstract

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.

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

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Figure 1. Exact and numerical values of the normalized moments

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

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

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

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

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

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

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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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