Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delay

Figures & data

Figure 1. Numerical Solution of Example 7.1, with N = 64, M = 80. (a) $\epsilon =2^{-4}$ (b) $\epsilon =2^{-14}$.

Figure 2. Numerical Solution of Example 7.2, with N = 64 = M. (a) $\epsilon =2^{-4}$ (b) $\epsilon =2^{-14}$.

Figure 3. Numerical Solution of Example 7.3, with N = 64. (a) $\epsilon =2^{-8}$.

Table 1. $E_{\epsilon }^{N,\mathrm{\Delta t}},P_{\epsilon }^{N,\mathrm{\Delta t}},E^{N,\mathrm{\Delta t}},P^{N,\mathrm{\Delta t}}$, for Example 7.1 on Bakhvalov–Shishkin mesh.

$\epsilon \downarrow$	N = 32	$N=64$	$N=128$	N = 256	N = 512
	M = 40	M = 80	$M=160$	$M=320$	$M=640$
Before Richardson extrapolation
$2^{-12}$	3.6393e−03	2.0741e−03	1.2601e−03	7.1707e−04	3.8423e−04
$2^{-14}$	3.6366e−03	2.0736e−03	1.2545e−03	7.1481e−04	3.8334e−04
$2^{-16}$	3.6359e−03	2.0735e−03	1.2531e−03	7.1422e−04	3.8309e−04
$2^{-18}$	3.6358e−03	2.0735e−03	1.2527e−03	7.1407e−04	3.8303e−04
$2^{-20}$	3.6357e−03	2.0735e−03	1.2527e−03	7.1403e−04	3.8301e−04
$2^{-22}$	3.6357e−03	2.0735e−03	1.2526e−03	7.1402e−04	3.8301e−04
$2^{-24}$	3.6357e−03	2.0735e−03	1.2526e−03	7.1402e−04	3.8301e−04
$2^{-26}$	3.6357e−03	2.0735e−03	1.2526e−03	7.1402e−04	3.8301e−04
$2^{-28}$	3.6357e−03	2.0735e−03	1.2526e−03	7.1402e−04	3.8301e−04
$2^{-30}$	3.6357e−03	2.0735e−03	1.2526e−03	7.1402e−04	3.8301e−04
$2^{-32}$	3.6357e−03	2.0735e−03	1.2526e−03	7.1402e−04	3.8301e−04
$E^{N,\mathrm{\Delta t}}$	3.6393e−03	2.0741e−03	1.2601e−03	7.1707e−04	3.8423e−04
$P^{N,\mathrm{\Delta t}}$	0.8112	0.7190	0.8133	0.9001	–
After Richardson extrapolation
$2^{-12}$	3.1195e−03	9.3315e−04	2.7076e−04	7.6430e−05	2.1047e−05
$2^{-14}$	3.1500e−03	9.4711e−04	2.7683e−04	7.8930e−05	2.2018e−05
$2^{-16}$	3.1578e−03	9.5072e−04	2.7845e−04	7.9637e−05	2.2325e−05
$2^{-18}$	3.1598e−03	9.5163e−04	2.7886e−04	7.9820e−05	2.2407e−05
$2^{-20}$	3.1602e−03	9.5186e−04	2.7896e−04	7.9866e−05	2.2428e−05
$2^{-22}$	3.1604e−03	9.5192e−04	2.7899e−04	7.9877e−05	2.2433e−05
$2^{-24}$	3.1604e−03	9.5193e−04	2.7899e−04	7.9880e−05	2.2434e−05
$2^{-26}$	3.1604e−03	9.5193e−04	2.7899e−04	7.9880e−05	2.2435e−05
$2^{-28}$	3.1604e−03	9.5194e−04	2.7899e−04	7.9881e−05	2.2435e−05
$2^{-30}$	3.1604e−03	9.5194e−04	2.7900e−04	7.9884e−05	2.2439e−05
$2^{-32}$	3.1604e−03	9.5195e−04	2.7899e−04	7.9874e−05	2.2441e−05
$E_{\mathrm{ext}}^{N,\mathrm{\Delta t}}$	3.1604e−03	9.5195e−04	2.7900e−04	7.9884e−05	2.2441e−05
$P_{\mathrm{ext}}^{N,\mathrm{\Delta t}}$	1.7312	1.7706	1.8043	1.8318	–

Table 2. $E_{\epsilon }^{N,\mathrm{\Delta t}},P_{\epsilon }^{N,\mathrm{\Delta t}},E^{N,\mathrm{\Delta t}},P^{N,\mathrm{\Delta t}}$, for Example 7.2 on Bakhvalov–Shishkin mesh,with N = M.

$\epsilon \downarrow$	N = 16	N = 32	N = 64	N = 128	N = 256	N = 512
Before Richardson extrapolation
$2^{-14}$	7.2870e−03	3.1938e−03	1.6513e−03	9.2133e−04	4.8183e−04	2.4447e−04
$2^{-16}$	7.3001e−03	3.2027e−03	1.6588e−03	9.2875e−04	4.8900e−04	2.4984e−04
$2^{-18}$	7.3033e−03	3.2050e−03	1.6607e−03	9.3064e−04	4.9087e−04	2.5167e−04
$2^{-20}$	7.3041e−03	3.2055e−03	1.6611e−03	9.3112e−04	4.9134e−04	2.5213e−04
$2^{-22}$	7.3043e−03	3.2057e−03	1.6613e−03	9.3124e−04	4.9146e−04	2.5225e−04
$2^{-24}$	7.3044e−03	3.2057e−03	1.6613e−03	9.3127e−04	4.9149e−04	2.5228e−04
$2^{-26}$	7.3044e−03	3.2057e−03	1.6613e−03	9.3128e−04	4.9150e−04	2.5229e−04
$2^{-28}$	7.3044e−03	3.2057e−03	1.6613e−03	9.3128e−04	4.9150e−04	2.5229e−04
$2^{-30}$	7.3044e−03	3.2057e−03	1.6613e−03	9.3128e−04	4.9150e−04	2.5229e−04
$E^{N,\mathrm{\Delta t}}$	7.3044e−03	3.2057e−03	1.6613e−03	9.3128e−04	4.9150e−04	2.5229e−04
$P^{N,\mathrm{\Delta t}}$	1.1881	0.9483	0.8350	0.9220	0.9621	–
After Richardson extrapolation
$2^{-14}$	3.1941e−03	1.0609e−03	3.2872e−04	9.7039e−05	2.7660e−05	7.6726e−06
$2^{-16}$	3.1988e−03	1.0635e−03	3.3004e−04	9.7655e−05	2.7941e−05	7.8011e−06
$2^{-18}$	3.2000e−03	1.0642e−03	3.3037e−04	9.7809e−05	2.8012e−05	7.8330e−06
$2^{-20}$	3.2003e−03	1.0644e−03	3.3046e−04	9.7847e−05	2.8029e−05	7.8410e−06
$2^{-22}$	3.2003e−03	1.0644e−03	3.3048e−04	9.7857e−05	2.8034e−05	7.8430e−06
$2^{-24}$	3.2003e−03	1.0644e−03	3.3048e−04	9.7859e−05	2.8035e−05	7.8435e−06
$2^{-26}$	3.2003e−03	1.0644e−03	3.3048e−04	9.7860e−05	2.8035e−05	7.8436e−06
$2^{-28}$	3.2003e−03	1.0644e−03	3.3049e−04	9.7860e−05	2.8035e−05	7.8437e−06
$2^{-30}$	3.2003e−03	1.0644e−03	3.3049e−04	9.7861e−05	2.8037e−05	7.8446e−06
$E_{\mathrm{ext}}^{N,\mathrm{\Delta t}}$	3.2003e−03	1.0644e−03	3.3049e−04	9.7861e−05	2.8037e−05	7.8446e−06
$P_{\mathrm{ext}}^{N,\mathrm{\Delta t}}$	1.5882	1.6874	1.7558	1.8034	1.8376	–

Table 3. $E_{\epsilon }^{N,\mathrm{\Delta t}},P_{\epsilon }^{N,\mathrm{\Delta t}},E^{N,\mathrm{\Delta t}},P^{N,\mathrm{\Delta t}}$, for Example 7.3 on Bakhvalov–Shishkin mesh,with N = M.

$\epsilon \downarrow$	N = 16	N = 32	N = 64	N = 128	N = 256	N = 512
Before Richardson extrapolation
$2^{-14}$	1.63540e−02	7.99370e−03	3.95110e−03	1.96410e−03	9.79190e−04	4.88880e−04
$2^{-16}$	1.63570e−02	7.99510e−03	3.95180e−03	1.96450e−03	9.79360e−04	4.88970e−04
$2^{-18}$	1.63570e−02	7.99550e−03	3.95200e−03	1.96450e−03	9.79410e−04	4.88990e−04
$2^{-20}$	1.63570e−02	7.99560e−03	3.95200e−03	1.96460e−03	9.79420e−04	4.88990e−04
$2^{-22}$	1.63570e−02	7.99560e−03	3.95200e−03	1.96460e−03	9.79420e−04	4.89000e−04
$2^{-24}$	1.63570e−02	7.99560e−03	3.95200e−03	1.96460e−03	9.79420e−04	4.89000e−04
$2^{-26}$	1.63570e−02	7.99560e−03	3.95200e−03	1.96460e−03	9.79420e−04	4.89000e−04
$2^{-28}$	1.63570e−02	7.99560e−03	3.95200e−03	1.96460e−03	9.79420e−04	4.89000e−04
$2^{-30}$	1.63570e−02	7.99560e−03	3.95200e−03	1.96460e−03	9.79410e−04	4.89000e−04
$E^{N,\mathrm{\Delta t}}$	1.63570e−02	7.99560e−03	3.95200e−03	1.96460e−03	9.79420e−04	4.89000e−04
$P^{N,\mathrm{\Delta t}}$	1.0327	1.0166	1.0084	1.0042	1.0021	–
After Richardson extrapolation
$2^{-20}$	1.1216e−03	3.2191e−04	8.5972e−05	2.2231e−05	5.6865e−06	1.4732e−06
$2^{-22}$	1.1215e−03	3.2188e−04	8.5933e−05	2.2192e−05	5.6464e−06	1.4328e−06
$2^{-24}$	1.1215e−03	3.2187e−04	8.5923e−05	2.2182e−05	5.6363e−06	1.4227e−06
$2^{-26}$	1.1215e−03	3.2187e−04	8.5921e−05	2.2179e−05	5.6338e−06	1.4202e−06
$2^{-28}$	1.1215e−03	3.2187e−04	8.5920e−05	2.2179e−05	5.6332e−06	1.4196e−06
$2^{-30}$	1.1215e−03	3.2186e−04	8.5920e−05	2.2179e−05	5.6330e−06	1.4194e−06
$E_{\mathrm{ext}}^{N,\mathrm{\Delta t}}$	1.1216e−03	3.2191e−04	8.5972e−05	2.2231e−05	5.6865e−06	1.4732e−06
$P_{\mathrm{ext}}^{N,\mathrm{\Delta t}}$	1.8008	1.9047	1.9513	1.967	1.9486	–

Table 4. $E_{\epsilon }^{N,\mathrm{\Delta t}},E^{N,\mathrm{\Delta t}}$, for Example 7.1 on Uniform mesh with ${\lambda }_1=1/12$ and ${\lambda }_2=5/12$.

$\epsilon \downarrow$	N = 16	N = 32	$N=64$	N = 128	$N=256$	N = 512
	M = 20	M = 40	M = 80	M = 160	M = 320	M = 640
$2^{-4}$	7.6851e−03	1.8001e−03	9.2424e−04	4.7309e−04	2.3985e−04	1.2081e−04
$2^{-8}$	4.1614e−01	2.4193e−01	1.1678e−01	3.9420e−02	9.4794e−03	2.0822e−03
$2^{-12}$	8.1993e−01	7.9584e−01	6.5128e−01	5.2501e−01	4.0498e−01	2.5430e−01
$2^{-16}$	8.6028e−01	9.1966e−01	9.3446e−01	8.8004e−01	7.4237e−01	6.6215e−01
$2^{-20}$	8.6288e−01	9.2858e−01	9.6553e−01	9.8053e−01	9.6949e−01	9.0382e−01
$2^{-24}$	8.6305e−01	9.2914e−01	9.6756e−01	9.8809e−01	9.9757e−01	9.9747e−01
$2^{-28}$	8.6306e−01	9.2917e−01	9.6769e−01	9.8857e−01	9.9941e−01	1.0047e+00
$E^{N,\mathrm{\Delta t}}$	8.6306e−01	9.2917e−01	9.6769e−01	9.8857e−01	9.9941e−01	1.0047e+00

Table 5. $E_{\epsilon }^{N,\mathrm{\Delta t}},E^{N,\mathrm{\Delta t}}$, for Example 7.2 on Uniform mesh with ${\lambda }_1=1/12$, ${\lambda }_2=5/12$ and $N=M$.

$\epsilon \downarrow$	N = 16	N = 32	$N=64$	$N=128$	N = 256	N = 512
$2^{-4}$	2.3921e−03	1.2926e−03	6.8897e−04	3.5867e−04	1.8315e−04	9.2628e−05
$2^{-8}$	1.0978e−01	7.7038e−02	3.8786e−02	1.3359e−02	3.2354e−03	6.9549e−04
$2^{-12}$	1.8478e−01	2.0408e−01	2.0053e−01	1.7868e−01	1.3950e−01	8.8087e−02
$2^{-16}$	1.9115e−01	2.2046e−01	2.3694e−01	2.4244e−01	2.3812e−01	2.2925e−01
$2^{-20}$	1.9155e−01	2.2153e−01	2.3967e−01	2.4946e−01	2.5378e−01	2.5361e−01
$2^{-24}$	1.9158e−01	2.2160e−01	2.3984e−01	2.4993e−01	2.5515e−01	2.5765e−01
$2^{-28}$	1.9158e−01	2.2160e−01	2.3985e−01	2.4996e−01	2.5524e−01	2.5794e−01
$E^{N,\mathrm{\Delta t}}$	1.9158e−01	2.2160e−01	2.3985e−01	2.4996e−01	2.5524e−01	2.5794e−01

Table 6. Comparison of $E_{\epsilon }^{N,\mathrm{\Delta t}}$ for Example 7.1 when N = M.

$\epsilon \downarrow$	N = 32	N = 64	N = 128	N = 256	N = 512
Proposed method
$2^{-6}$	4.4584e−03	2.9063e−03	1.6488e−03	9.4635e−04	8.8529e−04
$2^{-8}$	3.7809e−03	2.5402e−03	1.7113e−03	1.1520e−03	8.2689e−04
$2^{-10}$	3.8301e−03	2.1723e−03	1.3360e−03	7.8317e−04	4.9180e−04
$2^{-12}$	4.0055e−03	2.2524e−03	1.3159e−03	7.4830e−04	4.0089e−04
$2^{-14}$	4.0509e−03	2.2830e−03	1.3102e−03	7.4604e−04	4.0000e−04
$2^{-16}$	4.0623e−03	2.2916e−03	1.3088e−03	7.4545e−04	3.9975e−04
$2^{-18}$	4.0652e−03	2.2937e−03	1.3084e−03	7.4529e−04	3.9969e−04
$2^{-20}$	4.0659e−03	2.2943e−03	1.3083e−03	7.4526e−04	3.9967e−04
$2^{-24}$	4.0662e−03	2.2944e−03	1.3083e−03	7.4524e−04	3.9967e−04
$2^{-26}$	4.0662e−03	2.2945e−03	1.3083e−03	7.4524e−04	3.9967e−04
$2^{-30}$	4.0662e−03	2.2945e−03	1.3083e−03	7.4524e−04	3.9966e−04
Method in [57]
$2^{-6}$	3.3040e−03	1.1073e−03	4.1432e−04	1.7826e−04	8.4960e−05
$2^{-8}$	7.1097e−03	3.0887e−03	9.6542e−04	3.1847e−04	1.1859e−04
$2^{-10}$	7.2307e−03	3.8516e−03	1.9512e−03	8.0308e−04	2.5301e−04
$2^{-12}$	7.2307e−03	3.8523e−03	1.9892e−03	1.0105e−03	4.9936e−04
$2^{-14}$	7.2307e−03	3.8523e−03	1.9892e−03	1.0107e−03	5.0944e−04
$2^{-16}$	7.2307e−03	3.8523e−03	1.9892e−03	1.0107e−03	5.0944e−04
$2^{-18}$	7.2307e−03	3.8523e−03	1.9892e−03	1.0107e−03	5.0944e−04
$2^{-20}$	7.2307e−03	3.8523e−03	1.9892e−03	1.0107e−03	5.0944e−04
$2^{-24}$	7.2307e−03	3.8523e−03	1.9892e−03	1.0107e−03	5.0944e−04
$2^{-26}$	7.2307e−03	3.8523e−03	1.9892e−03	1.0107e−03	5.0944e−04
$2^{-30}$	7.2307e−03	3.8523e−03	1.9892e−03	1.0107e−03	5.0944e−04

Table 7. Comparison of $E_{\epsilon }^{N,\mathrm{\Delta t}}$ for Example 7.2 when N = M.

$\epsilon \downarrow$	N = 16	N = 32	N = 64	N = 128	N = 256	N = 512
Proposed method
$2^{-14}$	7.2870e−03	3.1938e−03	1.6513e−03	9.2133e−04	4.8183e−04	2.4447e−04
$2^{-16}$	7.3001e−03	3.2027e−03	1.6588e−03	9.2875e−04	4.8900e−04	2.4984e−04
$2^{-18}$	7.3033e−03	3.2050e−03	1.6607e−03	9.3064e−04	4.9087e−04	2.5167e−04
$2^{-20}$	7.3041e−03	3.2055e−03	1.6611e−03	9.3112e−04	4.9134e−04	2.5213e−04
$2^{-22}$	7.3043e−03	3.2057e−03	1.6613e−03	9.3124e−04	4.9146e−04	2.5225e−04
$2^{-24}$	7.3044e−03	3.2057e−03	1.6613e−03	9.3127e−04	4.9149e−04	2.5228e−04
$2^{-26}$	7.3044e−03	3.2057e−03	1.6613e−03	9.3128e−04	4.9150e−04	2.5229e−04
$2^{-28}$	7.3044e−03	3.2057e−03	1.6613e−03	9.3128e−04	4.9150e−04	2.5229e−04
Method in [58]
$2^{-14}$	4.9006e−02	2.8622e−02	1.5141e−02	7.7173e−03	3.8858e−03	1.9484e−03
$2^{-16}$	4.9006e−02	2.8622e−02	1.5141e−02	7.7173e−03	3.8858e−03	1.9484e−03
$2^{-18}$	4.9006e−02	2.8622e−02	1.5141e−02	7.7173e−03	3.8858e−03	1.9484e−03
$2^{-20}$	4.9006e−02	2.8622e−02	1.5141e−02	7.7173e−03	3.8858e−03	1.9484e−03
$2^{-22}$	4.9006e−02	2.8622e−02	1.5141e−02	7.7173e−03	3.8858e−03	1.9484e−03
$2^{-24}$	4.9006e−02	2.8622e−02	1.5141e−02	7.7173e−03	3.8858e−03	1.9484e−03
$2^{-26}$	4.9006e−02	2.8622e−02	1.5141e−02	7.7173e−03	3.8858e−03	1.9484e−03
$2^{-28}$	4.9006e−02	2.8622e−02	1.5141e−02	7.7173e−03	3.8858e−03	1.9484e−03

Figure 4. log–log plot of maximum point wise error for Example 7.1. (a) Before Extrapolation (b) After Extrapolation.

Figure 5. log–log plot of maximum point wise error for Example 7.2 (a) Before Extrapolation (b) After Extrapolation.

Figure 6. log–log plot of maximum point wise error for Example 7.3 (a) Before Extrapolation (b) After Extrapolation.

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