Computational method for singularly perturbed two-parameter parabolic convection-diffusion problems

Figures & data

Figure 1. Log-log plot error for Example 1.2 with $N$ number of space mesh and $\mu ={10}^{-7}$

Table 1. The maximum point-wise error and rate of convergence for example 1.1 where $\epsilon =2^{-5}$ and different values of $\mu$

$\mu \downarrow$	$N=32$	$N=64$	$N=128$	$N=256$	$N=512$
	$M=16$	$M=32$	$N=64$	$N=128$	$N=256$
Current method
$2^{-6}$	$2.4078e-04$	$6.0546e-05$	$1.5181e-05$	$3.7997e-06$	$9.5053e-07$
$2^{-8}$	$2.3648e-04$	$5.9390e-05$	$1.4884e-05$	$3.7252e-06$	$9.3180e-07$
$2^{-10}$	$2.3571e-04$	$5.9186e-05$	$1.4827e-05$	$3.7106e-06$	$9.2814e-07$
$2^{-12}$	$2.3551e-04$	$5.9135e-05$	$1.4814e-05$	$3.7073e-06$	$9.2729e-07$
$2^{-14}$	$2.3546e-04$	$5.9122e-05$	$1.4811e-05$	$3.7065e-06$	$9.2708e-07$
$\mathrm{P}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{a}$	$2.3545e-04$	$5.9119e-05$	$1.4810e-05$	$3.7063e-06$	$9.2703e-07$
$2^{-18}$	$2.3545e-04$	$5.9118e-05$	$1.4810e-05$	$3.7062e-06$	$9.2702e-07$
$2^{-20}$	$2.3545e-04$	$5.9118e-05$	$1.4810e-05$	$3.7062e-06$	$9.2702e-07$
$2^{-22}$	$2.3545e-04$	$5.9118e-05$	$1.4810e-05$	$3.7062e-06$	$9.2726e-07$
$2^{-24}$	$2.3544e-04$	$5.9118e-05$	$1.4810e-05$	$3.7062e-06$	$9.2713e-07$
$R$	1.9916	1.9958	1.9983	1.9991	1.9996
Method in Kadalbajoo and Yadaw (2012)
$2^{-6}$	$5.091e-3$	$4.580e-3$	$3.325e-3$	$2.004e-3$	$1.101e-3$
$2^{-8}$	$5.090e-3$	$4.327e-3$	$3.148e-3$	$1.898e-3$	$1.043e-3$
$2^{-10}$	$5.088e-3$	$4.265e-3$	$3.105e-3$	$1.873e-3$	$1.029e-3$
$2^{-12}$	$5.089e-3$	$4.250e-3$	$3.094e-3$	$1.866e-3$	$1.026e-3$
$2^{-14}$	$5.087e-3$	$4.246e-3$	$3.091e-3$	$1.865e-3$	$1.024e-3$
$2^{-16}$	$5.087e-3$	$4.245e-3$	$3.091e-3$	$1.864e-3$	$1.024e-3$
$2^{-18}$	$5.087e-3$	$4.245e-3$	$3.091e-3$	$1.864e-3$	$1.024e-3$
$2^{-20}$	$5.087e-3$	$4.245e-3$	$3.091e-3$	$1.864e-3$	$1.024e-3$
$2^{-22}$	$5.087e-3$	$4.245e-3$	$3.091e-3$	$1.864e-3$	$1.024e-3$
$2^{-24}$	$5.087e-3$	$4.245e-3$	$3.091e-3$	$1.864e-3$	$1.024e-3$
$R$	0.1527	0.4670	0.7302	0.8640	0.9314

Table 2. The maximum point-wise error and rate of convergence for example 1.2 where $\mu ={10}^{-7}$ and different values of $\epsilon$

$\epsilon \downarrow$	$N=64$	$N=128$	$N=256$	$N=512$
	$M=16$	$M=32$	$N=64$	$N=128$
Current method
${10}^{-6}$	$2.6377e-05$	$6.5877e-06$	$1.6474e-06$	$4.1182e-07$
${10}^{-7}$	$2.6377e-05$	$6.5877e-06$	$1.6474e-06$	$4.1182e-07$
${10}^{-8}$	$2.6377e-05$	$6.5877e-06$	$1.6474e-06$	$4.1182e-07$
${10}^{-9}$	$2.6377e-05$	$6.5877e-06$	$1.6474e-06$	$4.1182e-07$
$R$	$2.0014$	$1.9996$	$2.0001$	$1.9999$
Result in Gupta et al. (2019)
${10}^{-6}$	$3.8754e-5$	$1.0214e-5$	$2.6170e-6$	$6.6241e-7$
${10}^{-7}$	$3.8753e-5$	$1.0214e-5$	$2.6170e-6$	$6.6241e-7$
${10}^{-8}$	$3.8753e-5$	$1.0214e-5$	$2.6170e-6$	$6.6241e-7$
${10}^{-9}$	$3.8753e-5$	$1.0214e-5$	$2.5461e-6$	$6.6241e-7$
$R$	$1.9238$	$1.9646$	$1.9821$	$1.9910$
Result in Das and Mehrmann (2016)
${10}^{-6}$	$9.6949e-4$	$4.9906e-4$	$2.5231e-4$	$1.2824e-4$
${10}^{-7}$	$9.8712e-4$	$5.0049e-4$	$2.5485e-4$	$1.2853e-4$
${10}^{-8}$	$9.5128e-4$	$5.0026e-4$	$2.5237e-4$	$1.2781e-4$
${10}^{-9}$	$9.6746e-4$	$5.0012e-4$	$2.5461e-4$	$1.2804e-4$
$R$	$0.95193$	$0.97394$	$0.99186$	$0.99188$

Figure 4. Numerical solution of Example 1.1 for $N=M=2^9,\epsilon =2^{-24}$ and $\mu =2^{-8}$

Figure 2. Log-log plot error for Example 1.1 with $N$ number of space mesh and $\epsilon =2^{-5}$

Figure 3. Numerical solution of Example 1.2 for $N=2^9,M=2^8,\epsilon ={10}^{-9}$ and $\mu ={10}^{-7}$

Kadalbajoo, M., & Yadaw, A. S. (2012). Parameter-uniform finite element method for two-parameter singularly perturbed parabolic reaction- diffusion problems. International Journal of Computational Methods, 9(4), 1250047. https://doi.org/10.1142/S0219876212500478

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Gupta, V., Kadalbajoo, M. K., & Dubey, R. K. (2019). A parameter-uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters. International Journal of Computer Mathematics, 96(3), 474–499. https://doi.org/10.1080/00207160.2018.1432856

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Das, P., & Mehrmann, V. (2016). Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters. BIT Numerical Mathematics, 56(1), 51–76. https://doi.org/10.1007/s10543-015-0559-8

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