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Theory and Modern Applications

Table 7 The solution errors of Example 7 for \(\varepsilon=10^{-3}\) and \(\varepsilon=10^{-4}\) at \(h=0.001\)

From: Absolutely stable difference scheme for a general class of singular perturbation problems

\(x_{i}\)

ε = 10−3

ε = 10−4

CFD

ASCD

CFD

ASCD

 

\(\delta_{i}=\frac{4}{\omega|p_{i}|}\)

\(\delta_{i}=\delta_{\mathrm{op}}\)

 

\(\delta_{i}=\frac{4}{\omega|p_{i}|}\)

\(\delta_{i}=\delta_{\mathrm{op}}\)

0.00

0.0000e + 00

0.0000e + 00

0.0000e + 00

0.0000e + 00

0.0000e + 00

0.0000e + 00

0.01

2.5175e − 09

2.5175e − 09

2.5175e − 09

1.2904e − 01

1.0226e − 86

1.0226e − 86

0.02

9.4549e − 18

9.4549e − 18

9.4549e − 18

1.5959e − 02

5.7090e − 171

5.709e − 171

0.04

4.4278e − 34

4.4278e − 34

4.4278e − 34

2.1328e − 04

0.0000e + 00

0.0000e + 00

0.06

4.4806e − 50

1.0270e − 49

4.4806e − 50

2.3489e − 06

0.0000e + 00

0.0000e + 00

0.08

6.0646e − 50

1.1799e − 64

6.0646e − 50

2.0935e − 08

0.0000e + 00

0.0000e + 00

0.10

6.3669e − 50

6.7142e − 79

6.3669e − 50

1.4785e − 10

0.0000e + 00

0.0000e + 00

0.20

8.4790e − 50

1.0611e − 139

8.4790e − 50

2.8784e − 23

0.0000e + 00

0.0000e + 00

0.30

1.2675e − 49

3.9475e − 183

1.2675e − 49

2.5563e − 41

0.0000e + 00

0.0000e + 00

0.40

2.4911e − 49

3.4566e − 209

2.4911e − 49

2.8289e − 73

0.0000e + 00

0.0000e + 00

0.50

2.0204e − 48

7.1246e − 218

2.0204e − 48

8.4555e − 146

0.0000e + 00

0.0000e + 00

0.60

2.1996e − 39

3.4566e − 209

2.1996e − 39

1.3376e − 70

0.0000e + 00

0.0000e + 00

0.70

4.9323e − 21

3.9475e − 183

4.9323e − 21

2.5837e − 41

0.0000e + 00

0.0000e + 00

0.80

3.2994e − 21

1.0611e − 139

3.2994e − 21

2.9094e − 23

0.0000e + 00

0.0000e + 00

0.90

2.4776e − 21

6.7142e − 79

2.4776e − 21

1.4785e − 10

0.0000e + 00

0.0000e + 00

0.92

2.3599e − 21

1.1799e − 64

2.3599e − 21

2.0935e − 08

0.0000e + 00

0.0000e + 00

0.94

2.2529e − 21

1.0270e − 49

2.2529e − 21

2.3489e − 06

0.0000e + 00

0.0000e + 00

0.96

2.1552e − 21

4.4278e − 34

2.1552e − 21

2.1328e − 04

0.0000e + 00

0.0000e + 00

0.98

9.4528e − 18

9.4549e − 18

9.4528e − 18

1.5959e − 02

5.7090e − 171

5.709e − 171

0.99

2.5175e − 09

2.5175e − 09

2.5175e − 09

1.2904e − 01

1.0226e − 86

1.0226e − 86

1.00

0.0000e + 00

0.0000e + 00

0.0000e + 00

1.1102e − 16

0.0000e + 00

0.0000e + 00