Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation
Abstract
:1. Introduction
2. Heat Equation
2.1. Continuous Symmetry Groups
2.2. Discrete Symmetry Group
3. Finite Difference Schemes for the Heat Equation
3.1. Forward Difference Scheme
3.2. Backward Difference Scheme
3.3. Crank Nicolson Method
3.4. Invariantization of the Crank Nicolson Method under the Discrete Symmetry Transformation
4. Discrete Symmetry Numerical Scheme For the Heat Equation
5. Solutions of the Heat Equation
5.1. Analytic Solution
5.2. Numerical Solutions Using CNM and the Proposed Method DCNM
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
FTCS | Forward in Time and Centered in Space |
CNM | Crank Nicolson Method |
DCNM | Discritized Crank Nicolson Method |
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Generators | Symmetry Transformations | |
---|---|---|
1 | Space translation: | |
2 | Time translation: | |
3 | Scale Transformation: | |
4 | Scale Transformation: | |
5 | Galilean boost: | |
6 | Projection: | |
7 |
FTCS | CNM | DCNM | Exact Solutions | |
---|---|---|---|---|
0.0 | 0.000000 | 0.000000 | 0.0000000000 | 0.000000 |
0.2 | 0.000892 | 0.004517 | 0.0042956284 | 0.004227 |
0.4 | 0.001444 | 0.007309 | 0.00695047277 | 0.006840 |
0.6 | 0.001444 | 0.007309 | 0.00695047277 | 0.006840 |
0.8 | 0.000892 | 0.004517 | 0.0042956284 | 0.004227 |
1.0 | 0.000000 | 0.000000 | 0.0000000000 | 0.000000 |
FTCS | CNM | DCNM | |
---|---|---|---|
0.0 | 0.000000 | 0.000000 | 0.0000000000 |
0.2 | 0.003335 | ||
0.4 | 0.005396 | ||
0.6 | 0.005396 | ||
0.8 | 0.003335 | ||
1.0 | 0.000000 | 0.000000 | 0.000000000000 |
0.0 | 0.0000 | 0.0000 | 0.0000 |
0.2 | 0.0053 | 0.0045 | 0.0042 |
0.4 | 0.0085 | 0.0076 | 0.0069 |
0.6 | 0.0085 | 0.0076 | 0.0069 |
0.8 | 0.0053 | 0.0045 | 0.0042 |
1.0 | 0.0000 | 0.0000 | 0.0000 |
0.0 | 0.0000 | 0.0000 | 0.0000 |
0.2 | 0.0050 | 0.0047 | 0.0045 |
0.4 | 0.0081 | 0.0078 | 0.0076 |
0.6 | 0.0081 | 0.0078 | 0.0076 |
0.8 | 0.0050 | 0.0047 | 0.0045 |
1.0 | 0.0000 | 0.0000 | 0.0000 |
t | DCNM | DCNM | CNM | CNM |
---|---|---|---|---|
0.1 | ||||
0.3 | ||||
0.5 | ||||
0.7 | ||||
0.9 | ||||
1 |
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Bibi, K.; Feroze, T. Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation. Symmetry 2020, 12, 359. https://doi.org/10.3390/sym12030359
Bibi K, Feroze T. Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation. Symmetry. 2020; 12(3):359. https://doi.org/10.3390/sym12030359
Chicago/Turabian StyleBibi, Khudija, and Tooba Feroze. 2020. "Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation" Symmetry 12, no. 3: 359. https://doi.org/10.3390/sym12030359