Fitted numerical scheme for singularly perturbed parabolic differential- difference with time lag

Figures & data

Figure 1. 3D view of numerical solution for example 4.1 with $N=M=64,\delta =0.5\times \epsilon ,\eta =0.5\times \epsilon ,\xi =0.5\times \epsilon$.

Figure 2. (a), (b) and (c) effect of the $\epsilon ,\delta$ and η on the behaviour of the solution with layer formation when $N=64=M$, respectively.

Figure 3. Example 4.1 Log-Log scale plot of the maximum absolute error for different values of ɛ.

Table 1. $E_{\epsilon ,\delta ,\eta }^{N,\mathrm{\Delta }t},r_{\epsilon ,\delta ,\eta }^{N,\mathrm{\Delta }t},E^{N,\mathrm{\Delta }t}$and $r^{N,\mathrm{\Delta }t}$ for example 4.1 using the proposed method and results in (Sahu & Mohapatra, 2021) with $\delta =0.1\times \epsilon ,\eta =0.1\times \epsilon ,\xi =0.5\times \epsilon$

ɛ	N=16	32	64	128	256	512
$\downarrow$	M=8	16	32	64	128	256
Present Method
10^−3	8.6808$\times {10}^{-3}$	5.5873$\times {10}^{-3}$	3.1202$\times {10}^{-3}$	1.6719$\times {10}^{-3}$	8.3589$\times {10}^{-4}$	4.5042$\times {10}^{-4}$
	6.3568 $\times {10}^{-1}$	8.4051 $\times {10}^{-1}$	9.0015 $\times {10}^{-1}$	1.0001	8.9204 $\times {10}^{-1}$
10^−5	8.6852 $\times {10}^{-3}$	5.5901 $\times {10}^{-3}$	3.1225 $\times {10}^{-3}$	1.7041 $\times {10}^{-3}$	8.9350 $\times {10}^{-4}$	4.5725 $\times {10}^{-4}$
	6.3568 $\times {10}^{-1}$	8.4017 $\times {10}^{-1}$	8.7369 $\times {10}^{-1}$	9.3147 $\times {10}^{-1}$	9.6648 $\times {10}^{-1}$
10^−7	8.6852 $\times {10}^{-3}$	5.5901 $\times {10}^{-3}$	3.1225 $\times {10}^{-3}$	1.7041 $\times {10}^{-3}$	8.9350 $\times {10}^{-4}$	4.5725 $\times {10}^{-4}$
	6.3568 $\times {10}^{-1}$	8.4017 $\times {10}^{-1}$	8.7369 $\times {10}^{-1}$	9.3147 $\times {10}^{-1}$	9.6648 $\times {10}^{-1}$
10^−9	8.6852 $\times {10}^{-3}$	5.5901 $\times {10}^{-3}$	3.1225 $\times {10}^{-3}$	1.7041 $\times {10}^{-3}$	8.9350 $\times {10}^{-4}$	4.5725 $\times {10}^{-4}$
	6.3568 $\times {10}^{-1}$	8.4017$\times {10}^{-1}$	8.7369$\times {10}^{-1}$	9.3147$\times {10}^{-1}$	9.6648$\times {10}^{-1}$
$E^{N,\mathrm{\Delta }t}$	8.6852$\times {10}^{-3}$	5.5901$\times {10}^{-3}$	3.1225$\times {10}^{-3}$	1.7041$\times {10}^{-3}$	8.9350$\times {10}^{-4}$	4.5725$\times {10}^{-4}$
$r^{N,\mathrm{\Delta }t}$	6.3568$\times {10}^{-1}$	8.4017$\times {10}^{-1}$	8.7369$\times {10}^{-1}$	9.3147$\times {10}^{-1}$	9.6648$\times {10}^{-1}$
Results in (Sahu & Mohapatra, 2021)
10^−3	4.1480$\times {10}^{-2}$	2.8933$\times {10}^{-2}$	2.0069$\times {10}^{-2}$	1.3166$\times {10}^{-2}$	8.1760$\times {10}^{-3}$	4.8525$\times {10}^{-4}$
	0.5196	0.5277	0.6081	0.6873	0.7526
10^−5	4.1687$\times {10}^{-2}$	2.9042$\times {10}^{-2}$	2.0150$\times {10}^{-2}$	1.3216$\times {10}^{-2}$	8.2042$\times {10}^{-3}$	4.8694$\times {10}^{-3}$
	0.5214	0.5273	0.6084	0.6878	0.7526
10^−7	4.1689$\times {10}^{-2}$	2.9044$\times {10}^{-2}$	2.0151$\times {10}^{-2}$	1.3216$\times {10}^{-2}$	8.2045$\times {10}^{-3}$	4.8695$\times {10}^{-3}$
	0.5214	0.5273	0.6085	0.6878	0.7526
10^−9	4.1689$\times {10}^{-2}$	2.9044$\times {10}^{-2}$	2.0151$\times {10}^{-2}$	1.3216$\times {10}^{-2}$	8.2045$\times {10}^{-3}$	4.8695$\times {10}^{-3}$
	0.5214	0.5273	0.6085	0.6878	0.7526
$E^{N,\mathrm{\Delta }t}$	4.1689$\times {10}^{-2}$	2.9044$\times {10}^{-2}$	2.0151$\times {10}^{-2}$	1.3216$\times {10}^{-2}$	8.2045$\times {10}^{-3}$	4.8695$\times {10}^{-3}$
$r^{N,\mathrm{\Delta }t}$	0.5214	0.5273	0.6085	0.6878	0.7526

Table 2. $E_{\epsilon ,\delta ,\eta }^{N,\mathrm{\Delta }t},r_{\epsilon ,\delta ,\eta }^{N,\mathrm{\Delta }t},E^{N,\mathrm{\Delta }t}$and $r^{N,\mathrm{\Delta }t}$ for example 4.1 using the proposed method with $\epsilon ={10}^{-2},\xi =0.5\times \epsilon$

	N= 16	N=32	64	128	256	512
$\downarrow$	M=8	16	32	64	128	256
$\delta \downarrow$	$\eta =0.5\times \epsilon$
$0.1\times \epsilon$	8.0867$\times {10}^{-3}$	4.4416$\times {10}^{-3}$	2.6765$\times {10}^{-3}$	1.5293$\times {10}^{-3}$	8.2452$\times {10}^{-4}$	4.2998$\times {10}^{-4}$
	8.6447$\times {10}^{-1}$	7.3073$\times {10}^{-1}$	8.0748$\times {10}^{-1}$	8.9125$\times {10}^{-1}$	9.3928$\times {10}^{-1}$
$0.2\times \epsilon$	8.0782$\times {10}^{-3}$	4.4338$\times {10}^{-3}$	2.6713$\times {10}^{-3}$	1.5268$\times {10}^{-3}$	8.2302$\times {10}^{-4}$	4.2927$\times {10}^{-4}$
	8.6549$\times {10}^{-1}$	7.3100$\times {10}^{-1}$	8.0703$\times {10}^{-1}$	8.9151$\times {10}^{-1}$	9.3904$\times {10}^{-1}$
$0.3\times \epsilon$	8.0699$\times {10}^{-3}$	4.4259$\times {10}^{-3}$	2.6661$\times {10}^{-3}$	1.5242$\times {10}^{-3}$	8.2151$\times {10}^{-4}$	4.2856$\times {10}^{-4}$
	8.6658$\times {10}^{-1}$	7.3124$\times {10}^{-1}$	8.0668$\times {10}^{-1}$	8.9170$\times {10}^{-1}$	9.3878$\times {10}^{-1}$
$0.4\times \epsilon$	8.0624$\times {10}^{-3}$	4.4180$\times {10}^{-3}$	2.6609$\times {10}^{-3}$	1.5216$\times {10}^{-3}$	8.1998$\times {10}^{-4}$	4.2784$\times {10}^{-4}$
	8.6782$\times {10}^{-1}$	7.3148$\times {10}^{-1}$	8.0633$\times {10}^{-1}$	8.9193$\times {10}^{-1}$	9.3852$\times {10}^{-1}$
$0.5\times \epsilon$	8.0548$\times {10}^{-3}$	4.4101$\times {10}^{-3}$	2.6557$\times {10}^{-3}$	1.5189$\times {10}^{-3}$	8.1863$\times {10}^{-4}$	4.2711$\times {10}^{-4}$
	8.6904$\times {10}^{-1}$	7.3172$\times {10}^{-1}$	8.0607$\times {10}^{-1}$	8.9174$\times {10}^{-1}$	9.3860$\times {10}^{-1}$
$E^{N,\mathrm{\Delta }t}$	8.0867$\times {10}^{-3}$	4.4416$\times {10}^{-3}$	2.6765$\times {10}^{-3}$	1.5293$\times {10}^{-3}$	8.2452$\times {10}^{-4}$	4.2998$\times {10}^{-4}$
$r^{N,\mathrm{\Delta }t}$	8.6447$\times {10}^{-1}$	7.3073$\times {10}^{-1}$	8.0748$\times {10}^{-1}$	8.9125$\times {10}^{-1}$	9.3928$\times {10}^{-1}$
$\eta \downarrow$	$\delta =0.5\times \epsilon$
$0.1\times \epsilon$	8.0787$\times {10}^{-3}$	4.4351$\times {10}^{-3}$	2.6733$\times {10}^{-3}$	1.5271$\times {10}^{-3}$	8.2315$\times {10}^{-4}$	4.2930$\times {10}^{-4}$
	8.6516$\times {10}^{-1}$	7.3034$\times {10}^{-1}$	8.0783$\times {10}^{-1}$	8.9157$\times {10}^{-1}$	9.3917$\times {10}^{-1}$
$0.2\times \epsilon$	8.0727$\times {10}^{-3}$	4.4289$\times {10}^{-3}$	2.6689$\times {10}^{-3}$	1.5251$\times {10}^{-3}$	8.2195$\times {10}^{-4}$	4.2873$\times {10}^{-4}$
	8.6610$\times {10}^{-1}$	7.3070$\times {10}^{-1}$	8.0734$\times {10}^{-1}$	8.9178$\times {10}^{-1}$	9.3898$\times {10}^{-1}$
$0.3\times \epsilon$	8.0667$\times {10}^{-3}$	4.4226$\times {10}^{-3}$	2.6646$\times {10}^{-3}$	1.5230$\times {10}^{-3}$	8.2077$\times {10}^{-4}$	4.2818$\times {10}^{-4}$
	8.6708$\times {10}^{-1}$	7.3098$\times {10}^{-1}$	8.0700$\times {10}^{-1}$	8.9187$\times {10}^{-1}$	9.3876$\times {10}^{-1}$
$0.4\times \epsilon$	8.0607$\times {10}^{-3}$	4.4163$\times {10}^{-3}$	2.6601$\times {10}^{-3}$	1.5210$\times {10}^{-3}$	8.1962$\times {10}^{-4}$	4.2764$\times {10}^{-4}$
	8.6807$\times {10}^{-1}$	7.3136$\times {10}^{-1}$	8.0646$\times {10}^{-1}$	8.9199$\times {10}^{-1}$	9.3856$\times {10}^{-1}$
$0.5\times \epsilon$	8.0548$\times {10}^{-3}$	4.4101$\times {10}^{-3}$	2.6557$\times {10}^{-3}$	1.5189$\times {10}^{-3}$	8.1863$\times {10}^{-4}$	4.2711$\times {10}^{-4}$
	8.6904$\times {10}^{-1}$	7.3172$\times {10}^{-1}$	8.0607$\times {10}^{-1}$	8.9174$\times {10}^{-1}$	9.3860$\times {10}^{-1}$
$E^{N,\mathrm{\Delta }t}$	8.0787$\times {10}^{-3}$	4.4351$\times {10}^{-3}$	2.6733$\times {10}^{-3}$	1.5271$\times {10}^{-3}$	8.2315$\times {10}^{-4}$	4.2930$\times {10}^{-4}$
$r^{N,\mathrm{\Delta }t}$	8.6516$\times {10}^{-1}$	7.3034$\times {10}^{-1}$	8.0783$\times {10}^{-1}$	8.9157$\times {10}^{-1}$	9.3917$\times {10}^{-1}$

Sahu, S. R., & Mohapatra, J. (2021). Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shift. Journal of Applied Analysis, 28(1), 121–134. https://doi.org/10.1515/jaa-2021-2064

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Data availability statement

No data were used to support the study.