February  2019, 12(1): 79-103. doi: 10.3934/krm.2019004

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

Received  September 2017 Revised  April 2018 Published  July 2018

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.

Citation: Jitraj Saha, Nilima Das, Jitendra Kumar, Andreas Bück. Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinetic & Related Models, 2019, 12 (1) : 79-103. doi: 10.3934/krm.2019004
References:
[1]

E. Ben-Naim and P. L. Krapivsky, Multiscaling in fragmentation, Physica D: Nonlinear Phenomena, 107 (1997), 156-160.  doi: 10.1016/S0167-2789(97)00080-8.  Google Scholar

[2]

E. Bilgili and F. Capece, A rigorous breakage matrix methodology for characterization of multi-particle interactions in dense-phase particle breakage, Chemical Engineering Research and Design, 90 (2012), 1177-1188.  doi: 10.1016/j.cherd.2012.01.005.  Google Scholar

[3]

D. Boyer, G. Tarjus and P. Viot, Shattering transition in a multivariable fragmentation model, Physical Review E, 51 (1995), 1043. doi: 10.1103/PhysRevE.51.1043.  Google Scholar

[4]

D. BoyerG. Tarjus and P. Viot, Exact solution and multifractal analysis of a multivariable fragmentation model, Journal de Physique I, 7 (1997), 13-38.  doi: 10.1051/jp1:1997125.  Google Scholar

[5]

A. Buffo and V. Alopaeus, Solution of bivariate population balance equations with high-order moment-conserving method of classes, Computers & Chemical Engineering, 87 (2016), 111-124.  doi: 10.1016/j.compchemeng.2015.12.013.  Google Scholar

[6]

A. BuffoM. Jama and V. Alopaeus, Liquid-liquid extraction in a rotating disc column: Solution of 2d population balance with HMMC, Chemical Engineering Research and Design, 115 (2016), 270-281.  doi: 10.1016/j.cherd.2016.09.002.  Google Scholar

[7]

X. Deng and R. N. Davé, Breakage of fractal agglomerates, Chemical Engineering Science, 161 (2017), 117-126.  doi: 10.1016/j.ces.2016.12.018.  Google Scholar

[8]

M. DostaS. DaleS. AntonyukC. WassgrenS. Heinrich and J. D. Litster, Numerical and experimental analysis of influence of granule microstructure on ist compression breakage, Powder Technology, 299 (2016), 87-97.  doi: 10.1016/j.powtec.2016.05.005.  Google Scholar

[9]

H. Dündar and H. Benzer, Investigating multicomponent breakage in cement grinding, Minerals Engineering, 77 (2015), 131-136.  doi: 10.1016/j.mineng.2015.03.014.  Google Scholar

[10]

M. H. Ernst and G. Szamel, Fragmentation kinetics, Journal of Physics A: Mathematical and General, 26 (1993), 6085. doi: 10.1088/0305-4470/26/22/011.  Google Scholar

[11]

L. Forestier-Coste and S. Mancini, A finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence, SIAM Journal on Scientific Computing, 34 (2012), B840-B860.  doi: 10.1137/110847998.  Google Scholar

[12]

K. D. Kafui and C. Thornton, Numerical simulations of impact breakage of a spherical crystalline agglomerate, Powder Technology, 109 (2000), 113-132.  doi: 10.1016/S0032-5910(99)00231-4.  Google Scholar

[13]

G. KaurJ. Kumar and S. Heinrich, A weighted finite volume scheme for multivariate aggregation population balance equation, Computers & Chemical Engineering, 101 (2017), 1-10.  doi: 10.1016/j.compchemeng.2017.02.011.  Google Scholar

[14]

E. G. Kelly and D. J. Spottiswood, The breakage function; what is it really?, Minerals Engineering, 3 (1990), 405-414.  doi: 10.1016/0892-6875(90)90034-9.  Google Scholar

[15]

M. Kostoglou and A. G. Konstandopoulos, Evolution of aggregate size and fractal dimension during brownian coagulation, Journal of Aerosol Science, 32 (2001), 1399-1420.  doi: 10.1016/S0021-8502(01)00056-8.  Google Scholar

[16]

P. L. Krapivsky and E. Ben-Naim, Scaling and multiscaling in models of fragmentation, Physical Review E, 50 (1994), 3502. doi: 10.1103/PhysRevE.50.3502.  Google Scholar

[17]

J. KumarM. PeglowG. Warnecke and S. Heinrich, The cell average technique for solving multi-dimensional aggregation population balance equations, Computers & Chemical Engineering, 32 (2008), 1810-1830.  doi: 10.1016/j.compchemeng.2007.10.001.  Google Scholar

[18]

R. KumarJ. Kumar and G. Warnecke, Numerical methods for solving two-dimensional aggregation population balance equations, Computers & Chemical Engineering, 35 (2011), 999-1009.  doi: 10.1016/j.compchemeng.2010.08.002.  Google Scholar

[19]

A. MotaA. A. Vicente and J. Teixeira, Effect of spent grains on flow regime transition in bubble column, Chemical Engineering Science, 66 (2011), 3350-3357.  doi: 10.1016/j.ces.2011.01.042.  Google Scholar

[20]

M. N. Nandanwar and S. Kumar, A new discretization of space for the solution of multi-dimensional population balance equations, Chemical Engineering Science, 63 (2008), 2198-2210.  doi: 10.1016/j.ces.2008.01.015.  Google Scholar

[21]

M. N. Nandanwar and S. Kumar, A new discretization of space for the solution of multi-dimensional population balance equations: Simultaneous breakup and aggregation of particles, Chemical Engineering Science, 63 (2008), 3988-3997.  doi: 10.1016/j.ces.2008.04.054.  Google Scholar

[22]

E. PahijaY. ZhangM. WangY. Zhu and C. W. Hui, Microalgae growth determination using modified breakage equation model, Computer Aided Chemical Engineering, 37 (2015), 389-394.  doi: 10.1016/B978-0-444-63578-5.50060-8.  Google Scholar

[23]

S. Qamar and G. Warnecke, Solving population balance equations for two-component aggregation by a finite volume scheme, Chemical Engineering Science, 62 (2007), 679-693.  doi: 10.1016/j.ces.2006.10.001.  Google Scholar

[24]

S. QamarS. NoorQ. ul Ain and A. Seidel-Morgenstern, Bivariate extension of the quadrature method of moments for batch crystallization models, Industrial & Engineering Chemistry Research, 49 (2010), 11633-11644.  doi: 10.1021/ie101108s.  Google Scholar

[25]

D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic press, 2000. Google Scholar

[26]

G. J. Rodgers and M. K. Hassan, Fragmentation of particles with more than one degree of freedom, Physical Review E, 50 (1994), 3458. doi: 10.1103/PhysRevE.50.3458.  Google Scholar

[27]

D. E. RosnerR. McGraw and P. Tandon, Multivariate population balances via moment and Monte Carlo simulation methods: An important sol reaction engineering bivariate example and "mixed"? moments for the estimation of deposition, scavenging, and optical properties for populations of nonspherical suspended particles, Industrial & Engineering Chemistry Research, 42 (2003), 2699-2711.  doi: 10.1021/ie020627l.  Google Scholar

[28]

D. E. Rosner and S. Yu, MC simulation of aerosol aggregation and simultaneous spheroidization, AIChE journal, 47 (2001), 545-561.  doi: 10.1002/aic.690470305.  Google Scholar

[29]

J. Saha, J. Kumar, A. Bück and E. Tsotsas, Finite volume approximations of breakage population balance equation, Chemical Engineering Research and Design, 2016. doi: 10.1016/j.cherd.2016.02.012.  Google Scholar

[30]

W. SchubertM. Khanal and J. Tomas, Impact crushing of particle-particle compounds-experiment and simulation, International Journal of Mineral Processing, 75 (2005), 41-52.  doi: 10.1016/j.minpro.2004.01.006.  Google Scholar

[31]

M. SinghJ. KumarA. Bück and E. Tsotsas, An improved and efficient finite volume scheme for bivariate aggregation population balance equation, Journal of Computational and Applied Mathematics, 308 (2016), 83-97.  doi: 10.1016/j.cam.2016.04.037.  Google Scholar

[32]

P. Singh and M. K. Hassan, Kinetics of multidimensional fragmentation, Physical Review E, 53 (1996), 3134. doi: 10.1103/PhysRevE.53.3134.  Google Scholar

[33]

B. Tabis and R. Grzywacz, Numerical and technological properties of bubble column bioreactors for aerobic processes, Computers & Chemical Engineering, 35 (2011), 212-219.  doi: 10.1016/j.compchemeng.2010.03.015.  Google Scholar

[34]

P. Tandon and D. E. Rosner, Monte Carlo simulation of particle aggregation and simultaneous restructuring, Journal of Colloid and Interface Science, 213 (1999), 273-286.  doi: 10.1006/jcis.1998.6036.  Google Scholar

[35]

P. Toneva and W. Peukert, A general approach for the characterization of fragmentation problems, Advanced Powder Technology, 18 (2007), 39-51.  doi: 10.1163/156855207779768160.  Google Scholar

[36]

H. M. Vale and T. F. McKenna, Solution of the population balance equation for two-component aggregation by an extended fixed pivot technique, Industrial & engineering chemistry research, 44 (2005), 7885-7891.  doi: 10.1021/ie050179s.  Google Scholar

[37]

L. Vogel and W. Peukert, From single particle impact behaviour to modelling of impact mills, Chemical Engineering Science, 60 (2005), 5164-5176.  doi: 10.1016/j.ces.2005.03.064.  Google Scholar

[38]

X. WangW. GuiC. Yang and Y. Wang, Breakage distribution estimation of bauxite based on piecewise linearized breakage rate, Chinese Journal of Chemical Engineering, 20 (2012), 1198-1205.  doi: 10.1016/S1004-9541(12)60608-9.  Google Scholar

[39]

D. L. WrightR. McGraw and D. E. Rosner, Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering of particle populations, Journal of Colloid and Interface Science, 236 (2001), 242-251.  doi: 10.1006/jcis.2000.7409.  Google Scholar

[40]

F. XiaoH. XuX. Li and D. Wang, Modeling particle-size distribution dynamics in a shear-induced breakage process with an improved breakage kernel: Importance of the internal bonds, Colloids and Surfaces A: Physichemical Engineering Aspects, 468 (2015), 87-94.  doi: 10.1016/j.colsurfa.2014.11.060.  Google Scholar

show all references

References:
[1]

E. Ben-Naim and P. L. Krapivsky, Multiscaling in fragmentation, Physica D: Nonlinear Phenomena, 107 (1997), 156-160.  doi: 10.1016/S0167-2789(97)00080-8.  Google Scholar

[2]

E. Bilgili and F. Capece, A rigorous breakage matrix methodology for characterization of multi-particle interactions in dense-phase particle breakage, Chemical Engineering Research and Design, 90 (2012), 1177-1188.  doi: 10.1016/j.cherd.2012.01.005.  Google Scholar

[3]

D. Boyer, G. Tarjus and P. Viot, Shattering transition in a multivariable fragmentation model, Physical Review E, 51 (1995), 1043. doi: 10.1103/PhysRevE.51.1043.  Google Scholar

[4]

D. BoyerG. Tarjus and P. Viot, Exact solution and multifractal analysis of a multivariable fragmentation model, Journal de Physique I, 7 (1997), 13-38.  doi: 10.1051/jp1:1997125.  Google Scholar

[5]

A. Buffo and V. Alopaeus, Solution of bivariate population balance equations with high-order moment-conserving method of classes, Computers & Chemical Engineering, 87 (2016), 111-124.  doi: 10.1016/j.compchemeng.2015.12.013.  Google Scholar

[6]

A. BuffoM. Jama and V. Alopaeus, Liquid-liquid extraction in a rotating disc column: Solution of 2d population balance with HMMC, Chemical Engineering Research and Design, 115 (2016), 270-281.  doi: 10.1016/j.cherd.2016.09.002.  Google Scholar

[7]

X. Deng and R. N. Davé, Breakage of fractal agglomerates, Chemical Engineering Science, 161 (2017), 117-126.  doi: 10.1016/j.ces.2016.12.018.  Google Scholar

[8]

M. DostaS. DaleS. AntonyukC. WassgrenS. Heinrich and J. D. Litster, Numerical and experimental analysis of influence of granule microstructure on ist compression breakage, Powder Technology, 299 (2016), 87-97.  doi: 10.1016/j.powtec.2016.05.005.  Google Scholar

[9]

H. Dündar and H. Benzer, Investigating multicomponent breakage in cement grinding, Minerals Engineering, 77 (2015), 131-136.  doi: 10.1016/j.mineng.2015.03.014.  Google Scholar

[10]

M. H. Ernst and G. Szamel, Fragmentation kinetics, Journal of Physics A: Mathematical and General, 26 (1993), 6085. doi: 10.1088/0305-4470/26/22/011.  Google Scholar

[11]

L. Forestier-Coste and S. Mancini, A finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence, SIAM Journal on Scientific Computing, 34 (2012), B840-B860.  doi: 10.1137/110847998.  Google Scholar

[12]

K. D. Kafui and C. Thornton, Numerical simulations of impact breakage of a spherical crystalline agglomerate, Powder Technology, 109 (2000), 113-132.  doi: 10.1016/S0032-5910(99)00231-4.  Google Scholar

[13]

G. KaurJ. Kumar and S. Heinrich, A weighted finite volume scheme for multivariate aggregation population balance equation, Computers & Chemical Engineering, 101 (2017), 1-10.  doi: 10.1016/j.compchemeng.2017.02.011.  Google Scholar

[14]

E. G. Kelly and D. J. Spottiswood, The breakage function; what is it really?, Minerals Engineering, 3 (1990), 405-414.  doi: 10.1016/0892-6875(90)90034-9.  Google Scholar

[15]

M. Kostoglou and A. G. Konstandopoulos, Evolution of aggregate size and fractal dimension during brownian coagulation, Journal of Aerosol Science, 32 (2001), 1399-1420.  doi: 10.1016/S0021-8502(01)00056-8.  Google Scholar

[16]

P. L. Krapivsky and E. Ben-Naim, Scaling and multiscaling in models of fragmentation, Physical Review E, 50 (1994), 3502. doi: 10.1103/PhysRevE.50.3502.  Google Scholar

[17]

J. KumarM. PeglowG. Warnecke and S. Heinrich, The cell average technique for solving multi-dimensional aggregation population balance equations, Computers & Chemical Engineering, 32 (2008), 1810-1830.  doi: 10.1016/j.compchemeng.2007.10.001.  Google Scholar

[18]

R. KumarJ. Kumar and G. Warnecke, Numerical methods for solving two-dimensional aggregation population balance equations, Computers & Chemical Engineering, 35 (2011), 999-1009.  doi: 10.1016/j.compchemeng.2010.08.002.  Google Scholar

[19]

A. MotaA. A. Vicente and J. Teixeira, Effect of spent grains on flow regime transition in bubble column, Chemical Engineering Science, 66 (2011), 3350-3357.  doi: 10.1016/j.ces.2011.01.042.  Google Scholar

[20]

M. N. Nandanwar and S. Kumar, A new discretization of space for the solution of multi-dimensional population balance equations, Chemical Engineering Science, 63 (2008), 2198-2210.  doi: 10.1016/j.ces.2008.01.015.  Google Scholar

[21]

M. N. Nandanwar and S. Kumar, A new discretization of space for the solution of multi-dimensional population balance equations: Simultaneous breakup and aggregation of particles, Chemical Engineering Science, 63 (2008), 3988-3997.  doi: 10.1016/j.ces.2008.04.054.  Google Scholar

[22]

E. PahijaY. ZhangM. WangY. Zhu and C. W. Hui, Microalgae growth determination using modified breakage equation model, Computer Aided Chemical Engineering, 37 (2015), 389-394.  doi: 10.1016/B978-0-444-63578-5.50060-8.  Google Scholar

[23]

S. Qamar and G. Warnecke, Solving population balance equations for two-component aggregation by a finite volume scheme, Chemical Engineering Science, 62 (2007), 679-693.  doi: 10.1016/j.ces.2006.10.001.  Google Scholar

[24]

S. QamarS. NoorQ. ul Ain and A. Seidel-Morgenstern, Bivariate extension of the quadrature method of moments for batch crystallization models, Industrial & Engineering Chemistry Research, 49 (2010), 11633-11644.  doi: 10.1021/ie101108s.  Google Scholar

[25]

D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic press, 2000. Google Scholar

[26]

G. J. Rodgers and M. K. Hassan, Fragmentation of particles with more than one degree of freedom, Physical Review E, 50 (1994), 3458. doi: 10.1103/PhysRevE.50.3458.  Google Scholar

[27]

D. E. RosnerR. McGraw and P. Tandon, Multivariate population balances via moment and Monte Carlo simulation methods: An important sol reaction engineering bivariate example and "mixed"? moments for the estimation of deposition, scavenging, and optical properties for populations of nonspherical suspended particles, Industrial & Engineering Chemistry Research, 42 (2003), 2699-2711.  doi: 10.1021/ie020627l.  Google Scholar

[28]

D. E. Rosner and S. Yu, MC simulation of aerosol aggregation and simultaneous spheroidization, AIChE journal, 47 (2001), 545-561.  doi: 10.1002/aic.690470305.  Google Scholar

[29]

J. Saha, J. Kumar, A. Bück and E. Tsotsas, Finite volume approximations of breakage population balance equation, Chemical Engineering Research and Design, 2016. doi: 10.1016/j.cherd.2016.02.012.  Google Scholar

[30]

W. SchubertM. Khanal and J. Tomas, Impact crushing of particle-particle compounds-experiment and simulation, International Journal of Mineral Processing, 75 (2005), 41-52.  doi: 10.1016/j.minpro.2004.01.006.  Google Scholar

[31]

M. SinghJ. KumarA. Bück and E. Tsotsas, An improved and efficient finite volume scheme for bivariate aggregation population balance equation, Journal of Computational and Applied Mathematics, 308 (2016), 83-97.  doi: 10.1016/j.cam.2016.04.037.  Google Scholar

[32]

P. Singh and M. K. Hassan, Kinetics of multidimensional fragmentation, Physical Review E, 53 (1996), 3134. doi: 10.1103/PhysRevE.53.3134.  Google Scholar

[33]

B. Tabis and R. Grzywacz, Numerical and technological properties of bubble column bioreactors for aerobic processes, Computers & Chemical Engineering, 35 (2011), 212-219.  doi: 10.1016/j.compchemeng.2010.03.015.  Google Scholar

[34]

P. Tandon and D. E. Rosner, Monte Carlo simulation of particle aggregation and simultaneous restructuring, Journal of Colloid and Interface Science, 213 (1999), 273-286.  doi: 10.1006/jcis.1998.6036.  Google Scholar

[35]

P. Toneva and W. Peukert, A general approach for the characterization of fragmentation problems, Advanced Powder Technology, 18 (2007), 39-51.  doi: 10.1163/156855207779768160.  Google Scholar

[36]

H. M. Vale and T. F. McKenna, Solution of the population balance equation for two-component aggregation by an extended fixed pivot technique, Industrial & engineering chemistry research, 44 (2005), 7885-7891.  doi: 10.1021/ie050179s.  Google Scholar

[37]

L. Vogel and W. Peukert, From single particle impact behaviour to modelling of impact mills, Chemical Engineering Science, 60 (2005), 5164-5176.  doi: 10.1016/j.ces.2005.03.064.  Google Scholar

[38]

X. WangW. GuiC. Yang and Y. Wang, Breakage distribution estimation of bauxite based on piecewise linearized breakage rate, Chinese Journal of Chemical Engineering, 20 (2012), 1198-1205.  doi: 10.1016/S1004-9541(12)60608-9.  Google Scholar

[39]

D. L. WrightR. McGraw and D. E. Rosner, Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering of particle populations, Journal of Colloid and Interface Science, 236 (2001), 242-251.  doi: 10.1006/jcis.2000.7409.  Google Scholar

[40]

F. XiaoH. XuX. Li and D. Wang, Modeling particle-size distribution dynamics in a shear-induced breakage process with an improved breakage kernel: Importance of the internal bonds, Colloids and Surfaces A: Physichemical Engineering Aspects, 468 (2015), 87-94.  doi: 10.1016/j.colsurfa.2014.11.060.  Google Scholar

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
$1$ $1$ $\frac{2}{y_1y_2}$ $\mathcal{M}_{k,l}(t) = \exp\left[\left(\frac{2}{(k+1)(l+1)} - 1\right)t\right]$
$2$ $1$ $2\delta\left(x_1-\frac{y_1}{2}\right)\delta\left(x_2-\frac{y_2}{2}\right)$ $\mathcal{M}_{k,l}(t) = \exp\left[\left(2^{1-k-l} - 1\right)t\right]$
$3$ $x_1+x_2$ $\frac{2}{y_1y_2}$ $\mathcal{M}_{1,0}(t) = \mathcal{M}_{0,1}(t) = 1$, $\mathcal{M}_{0,0} (t) = 1 + 2t$
$4$ $x_1+x_2$ $2\delta\left(x_1-\frac{y_1}{2}\right)\delta\left(x_2-\frac{y_2}{2}\right)$ $\mathcal{M}_{1,0}(t) = \mathcal{M}_{0,1}(t) = 1$, $\mathcal{M}_{0,0} (t) = 1 + 2t$
$5$ $1$ $\frac{4}{y_1y_2}$ $\mathcal{M}_{k,l}(t) = \exp\left[\left(\frac{4}{(k+1)(l+1)} - 1\right)t\right]$
$6$ $1$ $\frac{y_1\delta\left(x_1-y_1\right) + y_2\delta\left(x_2-y_2\right)}{y_1y_2}$ $\mathcal{M}_{1,1} (t) = 1$, $\mathcal{M}_{0,0} (t) = \exp(t)$, $\mathcal{M}_{1,0}(t) + \mathcal{M}_{0,1}(t) =\exp(t/2)$
$7$ $x_1+x_2$ $\frac{4}{y_1y_2}$ $\mathcal{M}_{1,1}(t) = 1$, $\mathcal{M}_{0,0}(t) = 1 + 3t$
$8$ $x_1+x_2$ $\frac{y_1\delta\left(x_1-y_1\right) + y_2\delta\left(x_2-y_2\right)}{y_1y_2}$ $\mathcal{M}_{1,1}(t) = 1$, $\mathcal{M}_{0,0}(t) = 1 + t$
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
$9$ $x_1+x_2+x_3$ $2\delta\left(x_1-\frac{y_1}{2}\right)\delta\left(x_2-\frac{y_2}{2}\right) \delta\left(x_3-\frac{y_3}{2}\right)$ $\mathcal{M}_{1,0,0}(t) = \mathcal{M}_{1,0,1}(t) = 1$, $\mathcal{M}_{0,0,1}(t) = 1$, $\mathcal{M}_{0,0,0}(t) = 1+ 3t$
$10$ $x_1x_2x_3$ $\frac{8}{y_1y_2y_3}$ $\mathcal{M}_{1,1,1}(t) = 1$, $\mathcal{M}_{0,0,0}(t) = 1+ 7t$
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Test case $9$ Test case $10$
Scheme$-1a$ Scheme$-2a$ Scheme$-1b$ Scheme$-2b$
58 86 13 26
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