A robust collocation method for singularly perturbed discontinuous coefficients parabolic differential difference equations

Figures & data

Table 1. Values of ${\mathcal{P}}_n(\mathit{s},\mathit{\delta })$ and its derivatives at the mesh points

s	$s_{\mathit{n}-2}$	$s_{\mathit{n}-1}$	sn	$s_{\mathit{n}+1}$	$s_{\mathit{n}+2}$
${\mathcal{P}}_n(\mathit{s},\mathit{\delta })$	0	${\displaystyle \frac{4-\mathit{\delta }}{24}}$	${\displaystyle \frac{8+\mathit{\delta }}{12}}$	${\displaystyle \frac{4-\mathit{\delta }}{24}}$	0
${\mathcal{P}}_n^{\prime }(\mathit{s},\mathit{\delta })$	0	$-{\displaystyle \frac{1}{2\mathit{h}}}$	0	${\displaystyle \frac{1}{2\mathit{h}}}$	0
${\mathcal{P}}_n^{\mathit{\prime \prime }}(\mathit{s},\mathit{\delta })$	0	${\displaystyle \frac{2+\mathit{\delta }}{2h^2}}$	$-{\displaystyle \frac{2+\mathit{\delta }}{h^2}}$	${\displaystyle \frac{2+\mathit{\delta }}{2h^2}}$	0

Table 2. Values of $E_{\epsilon }^{\mathit{N},\mathit{M}}$, $E^{\mathit{N},\mathit{M}}$ and $R^{\mathit{N},\mathit{M}}$ for example 5.1 when δ = 0.99 and N = M

$\downarrow \mathit{\epsilon }$	N = 32	N = 64	N = 128	N = 256	N = 512
10^−3	3.4892$\times {10}^{-3}$	1.8881$\times {10}^{-3}$	9.8074$\times {10}^{-4}$	4.9964$\times {10}^{-4}$	2.5215$\times {10}^{-4}$
10^−4	3.4892$\times {10}^{-3}$	1.8881$\times {10}^{-3}$	9.8074$\times {10}^{-4}$	4.9964$\times {10}^{-4}$	2.5215$\times {10}^{-4}$
10^−5	3.4892$\times {10}^{-3}$	1.8881$\times {10}^{-3}$	9.8074$\times {10}^{-4}$	4.9964$\times {10}^{-4}$	2.5215$\times {10}^{-4}$
10^−6	3.4892$\times {10}^{-3}$	1.8881$\times {10}^{-3}$	9.8074$\times {10}^{-4}$	4.9964$\times {10}^{-4}$	2.5215$\times {10}^{-4}$
10^−7	3.4892$\times {10}^{-3}$	1.8881$\times {10}^{-3}$	9.8074$\times {10}^{-4}$	4.9964$\times {10}^{-4}$	2.5215$\times {10}^{-4}$
10^−8	3.4892$\times {10}^{-3}$	1.8881$\times {10}^{-3}$	9.8074$\times {10}^{-4}$	4.9964$\times {10}^{-4}$	2.5215$\times {10}^{-4}$
10^−9	3.4892$\times {10}^{-3}$	1.8881$\times {10}^{-3}$	9.8074$\times {10}^{-4}$	4.9964$\times {10}^{-4}$	2.5215$\times {10}^{-4}$
10^−10	3.4892$\times {10}^{-3}$	1.8881$\times {10}^{-3}$	9.8074$\times {10}^{-4}$	4.9964$\times {10}^{-4}$	2.5215$\times {10}^{-4}$
${\mathbf{E}}^{\mathbf{N}\mathbf{,}\mathbf{M}}$	3.4892$\times {\mathbf{10}}^{\mathbf{-}\mathbf{3}}$	1.8881$\times {\mathbf{10}}^{\mathbf{-}\mathbf{3}}$	9.8074$\times {\mathbf{10}}^{\mathbf{-}\mathbf{4}}$	4.9964$\times {\mathbf{10}}^{\mathbf{-}\mathbf{4}}$	2.5215$\times {\mathbf{10}}^{\mathbf{-}\mathbf{4}}$
${\mathbf{R}}^{\mathbf{N}\mathbf{,}\mathbf{M}}$	8.8594$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	9.4500$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	9.7298$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	9.8661$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	-

Table 3. Values of $E_{\epsilon }^{\mathit{N},\mathit{M}}$, $E^{\mathit{N},\mathit{M}}$ and $R^{\mathit{N},\mathit{M}}$ for example 5.2 when δ = 0.99 and N = M

$\downarrow \mathit{\epsilon }$	N = 32	N = 64	N = 128	N = 256	N = 512
10^−3	9.8104$\times {10}^{-3}$	6.8163$\times {10}^{-3}$	4.6462$\times {10}^{-3}$	3.1734$\times {10}^{-3}$	2.1811$\times {10}^{-3}$
10^−4	9.8104$\times {10}^{-3}$	6.8163$\times {10}^{-3}$	4.6462$\times {10}^{-3}$	3.1734$\times {10}^{-3}$	2.1811$\times {10}^{-3}$
10^−5	9.8104$\times {10}^{-3}$	6.8163$\times {10}^{-3}$	4.6462$\times {10}^{-3}$	3.1734$\times {10}^{-3}$	2.1811$\times {10}^{-3}$
10^−6	9.8104$\times {10}^{-3}$	6.8163$\times {10}^{-3}$	4.6462$\times {10}^{-3}$	3.1734$\times {10}^{-3}$	2.1811$\times {10}^{-3}$
10^−7	9.8104$\times {10}^{-3}$	6.8163$\times {10}^{-3}$	4.6462$\times {10}^{-3}$	3.1734$\times {10}^{-3}$	2.1811$\times {10}^{-3}$
10^−8	9.8104$\times {10}^{-3}$	6.8163$\times {10}^{-3}$	4.6462$\times {10}^{-3}$	3.1734$\times {10}^{-3}$	2.1811$\times {10}^{-3}$
10^−9	9.8104$\times {10}^{-3}$	6.8163$\times {10}^{-3}$	4.6462$\times {10}^{-3}$	3.1734$\times {10}^{-3}$	2.1811$\times {10}^{-3}$
10^−10	9.8104$\times {10}^{-3}$	6.8163$\times {10}^{-3}$	4.6462$\times {10}^{-3}$	3.1734$\times {10}^{-3}$	2.1811$\times {10}^{-3}$
${\mathbf{E}}^{\mathbf{N}\mathbf{,}\mathbf{M}}$	9.8104$\times {10}^{-3}$	6.8163$\times {10}^{-3}$	4.6462$\times {10}^{-3}$	3.1734$\times {10}^{-3}$	2.1811$\times {10}^{-3}$
${\mathbf{R}}^{\mathbf{N}\mathbf{,}\mathbf{M}}$	5.2532$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	5.5293$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	5.5002$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	5.4101$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	-

Table 4. Values of $E_{\epsilon }^{\mathit{N},\mathit{M}}$, $E^{\mathit{N},\mathit{M}}$ and $R^{\mathit{N},\mathit{M}}$ for example 5.3 when δ = 0.99 and N = M

$\downarrow \mathit{\epsilon }$	N = 32	N = 64	N = 128	N = 256	N = 512
10^−3	3.6169$\times {10}^{-1}$	1.9086$\times {10}^{-1}$	9.8107$\times {10}^{-2}$	4.9616$\times {10}^{-2}$	2.9033$\times {10}^{-2}$
10^−4	3.6169$\times {10}^{-1}$	1.9086$\times {10}^{-1}$	9.8107$\times {10}^{-2}$	4.9740$\times {10}^{-2}$	2.9106$\times {10}^{-2}$
10^−5	3.6169$\times {10}^{-1}$	1.9086$\times {10}^{-1}$	9.8107$\times {10}^{-2}$	4.9740$\times {10}^{-2}$	2.9106$\times {10}^{-2}$
10^−6	3.6169$\times {10}^{-1}$	1.9086$\times {10}^{-1}$	9.8107$\times {10}^{-2}$	4.9740$\times {10}^{-2}$	2.9106$\times {10}^{-2}$
10^−7	3.6169$\times {10}^{-1}$	1.9086$\times {10}^{-1}$	9.8107$\times {10}^{-2}$	4.9740$\times {10}^{-2}$	2.9106$\times {10}^{-2}$
10^−8	3.6169$\times {10}^{-1}$	1.9086$\times {10}^{-1}$	9.8107$\times {10}^{-2}$	4.9740$\times {10}^{-2}$	2.9106$\times {10}^{-2}$
10^−9	3.6169$\times {10}^{-1}$	1.9086$\times {10}^{-1}$	9.8107$\times {10}^{-2}$	4.9740$\times {10}^{-2}$	2.9106$\times {10}^{-2}$
10^−10	3.6169$\times {10}^{-1}$	1.9086$\times {10}^{-1}$	9.8107$\times {10}^{-2}$	4.9740$\times {10}^{-2}$	2.9106$\times {10}^{-2}$
${\mathbf{E}}^{\mathbf{N}\mathbf{,}\mathbf{M}}$	3.6169$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	1.9086$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	9.8107$\times {10}^{-2}$	4.9740$\times {10}^{-2}$	2.9106$\times {10}^{-2}$
${\mathbf{R}}^{\mathbf{N}\mathbf{,}\mathbf{M}}$	9.2224$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	9.6010$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	9.7996$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	7.7309$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	-

Table 5. Comparison of ɛ-uniform values using the proposed method for example 5.1 when δ = 0.99 and N = M

$\mathit{\epsilon }={10}^{-10}$	N = 32	N = 64	N = 128	N = 256	N = 512
Results of the presented method
${\mathbf{E}}^{\mathbf{N}\mathbf{,}\mathbf{M}}$	3.4892$\times {10}^{-3}$	1.8881$\times {10}^{-3}$	9.8074$\times {10}^{-4}$	4.9964$\times {10}^{-4}$	2.5215$\times {10}^{-4}$
${\mathbf{R}}^{\mathbf{N}\mathbf{,}\mathbf{M}}$	8.8594$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	9.4500$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	9.7298$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	9.8661$\times {\mathbf{10}}^{\mathbf{-}\mathbf{1}}$	-
Results in (Ayele et al., 2022)
${\mathbf{E}}^{\mathbf{N}\mathbf{,}\mathbf{M}}$	3.6703$\times {10}^{-3}$	1.9480$\times {10}^{-3}$	1.0030$\times {10}^{-3}$	5.0885$\times {10}^{-4}$	2.5627$\times {10}^{-4}$
${\mathbf{R}}^{\mathbf{N}\mathbf{,}\mathbf{M}}$	0.9139	0.9577	0.9790	0.9896	-

Figure 1. Numerical solutions of example 5.1 for δ = 0.99 when $\mathit{N}=\mathit{M}=2^6$.

Figure 2. Numerical solutions of example 5.2 for δ = 0.99 when $\mathit{N}=\mathit{M}=2^6$.

Figure 3. Numerical solutions of example 5.3 for δ = 0.99 when $\mathit{N}=\mathit{M}=2^6$.

Figure 4. Line graphs at t = 2 when δ = 0.99 and $\mathit{N}=\mathit{M}=2^7$.

Figure 5. The log-log plot of the maximum absolute errors.

Ayele, M. A., Tiruneh, A. A., Derese, G. A., & Araci, S. (2022). Uniformly convergent scheme for singularly perturbed space delay parabolic differential equation with discontinuous convection coefficient and source term. Journal of Mathematics, 2022, 1–15. https://doi.org/10.1155/2022/1874741

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