# American Institute of Mathematical Sciences

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
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##### References:
Exact and numerical values of the normalized moments
Exact and numerical values of the normalized moments with size independent selection function
Exact and numerical values of the normalized moments with size dependent selection function
Exact and numerical values of the normalized moments with size independent selection function
Exact and numerical values of the normalized moments with size dependent selection function
Exact and numerical values of the normalized moments with the kernels having three particle properties
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 $1$ $1$ $\frac{2}{y_1y_2}$ $\mathcal{M}_{k,l}(t) = \exp\left[\left(\frac{2}{(k+1)(l+1)} - 1\right)t\right]$ $2$ $1$ $2\delta\left(x_1-\frac{y_1}{2}\right)\delta\left(x_2-\frac{y_2}{2}\right)$ $\mathcal{M}_{k,l}(t) = \exp\left[\left(2^{1-k-l} - 1\right)t\right]$ $3$ $x_1+x_2$ $\frac{2}{y_1y_2}$ $\mathcal{M}_{1,0}(t) = \mathcal{M}_{0,1}(t) = 1$, $\mathcal{M}_{0,0} (t) = 1 + 2t$ $4$ $x_1+x_2$ $2\delta\left(x_1-\frac{y_1}{2}\right)\delta\left(x_2-\frac{y_2}{2}\right)$ $\mathcal{M}_{1,0}(t) = \mathcal{M}_{0,1}(t) = 1$, $\mathcal{M}_{0,0} (t) = 1 + 2t$ $5$ $1$ $\frac{4}{y_1y_2}$ $\mathcal{M}_{k,l}(t) = \exp\left[\left(\frac{4}{(k+1)(l+1)} - 1\right)t\right]$ $6$ $1$ $\frac{y_1\delta\left(x_1-y_1\right) + y_2\delta\left(x_2-y_2\right)}{y_1y_2}$ $\mathcal{M}_{1,1} (t) = 1$, $\mathcal{M}_{0,0} (t) = \exp(t)$, $\mathcal{M}_{1,0}(t) + \mathcal{M}_{0,1}(t) =\exp(t/2)$ $7$ $x_1+x_2$ $\frac{4}{y_1y_2}$ $\mathcal{M}_{1,1}(t) = 1$, $\mathcal{M}_{0,0}(t) = 1 + 3t$ $8$ $x_1+x_2$ $\frac{y_1\delta\left(x_1-y_1\right) + y_2\delta\left(x_2-y_2\right)}{y_1y_2}$ $\mathcal{M}_{1,1}(t) = 1$, $\mathcal{M}_{0,0}(t) = 1 + t$
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 $9$ $x_1+x_2+x_3$ $2\delta\left(x_1-\frac{y_1}{2}\right)\delta\left(x_2-\frac{y_2}{2}\right) \delta\left(x_3-\frac{y_3}{2}\right)$ $\mathcal{M}_{1,0,0}(t) = \mathcal{M}_{1,0,1}(t) = 1$, $\mathcal{M}_{0,0,1}(t) = 1$, $\mathcal{M}_{0,0,0}(t) = 1+ 3t$ $10$ $x_1x_2x_3$ $\frac{8}{y_1y_2y_3}$ $\mathcal{M}_{1,1,1}(t) = 1$, $\mathcal{M}_{0,0,0}(t) = 1+ 7t$
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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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