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Theory and Modern Applications

Table 2 The solution errors of Example 2 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

 

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

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

 

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

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

0.000

3.6835e − 13

0.0000e + 00

3.6835e − 13

1.3312e − 12

0.0000e + 00

0.0000e + 00

0.001

3.4477e − 02

3.6814e − 01

3.4477e − 02

6.6658e − 01

7.5381e − 04

7.5381e − 04

0.002

2.4176e − 02

1.3606e − 01

2.4176e − 02

4.4436e − 01

7.9840e − 04

7.9840e − 04

0.003

1.2725e − 02

5.0684e − 02

1.2725e − 02

2.9624e − 01

7.9760e − 04

7.9760e − 04

0.004

5.9580e − 03

1.9275e − 02

5.9580e − 03

1.9749e − 01

7.9680e − 04

7.9680e − 04

0.005

2.6175e − 03

7.7195e − 03

2.6175e − 03

1.3166e − 01

7.9600e − 04

7.9600e − 04

0.006

1.1048e − 03

3.4678e − 03

1.1048e − 03

8.7774e − 02

7.9520e − 04

7.9520e − 04

0.007

4.5373e − 04

1.9031e − 03

4.5373e − 04

5.8516e − 02

7.9440e − 04

7.9440e − 04

0.008

1.8268e − 04

1.3268e − 03

1.8268e − 04

3.9011e − 02

7.9360e − 04

7.9360e − 04

0.009

7.2459e − 05

1.1142e − 03

7.2459e − 05

2.6007e − 02

7.9280e − 04

7.9280e − 04

0.010

2.8408e − 05

1.0353e − 03

2.8408e − 05

1.7338e − 02

7.9200e − 04

7.9200e − 04

0.020

1.7708e − 09

9.8000e − 04

1.7708e − 09

3.0067e − 04

7.8400e − 04

7.8400e − 04

0.050

5.5511e − 16

9.5000e − 04

5.5511e − 16

1.5680e − 09

7.6000e − 04

7.6000e − 04

0.100

4.4409e − 16

9.0000e − 04

4.4409e − 16

8.8818e − 16

7.2000e − 04

7.2000e − 04

0.300

2.2204e − 16

7.0000e − 04

2.2204e − 16

3.3307e − 16

5.6000e − 04

5.6000e − 04

0.500

2.2204e − 16

5.0000e − 04

2.2204e − 16

3.3307e − 16

4.0000e − 04

4.0000e − 04

1.000

0.0000e + 00

0.0000e + 00

0.0000e + 00

0.0000e + 00

0.0000e + 00

0.0000e + 00