Numerical Investigation of Heat Conduction with Multiple Moving Heat Sources
Abstract
:1. Introduction
- Firstly, the adaptive mesh is optimal in the sense of equidistribution on each subdomain, and can be generated very efficiently, since the local equidistributed mesh on each subdomain can be obtained in parallel.
- Secondly, it is found that the jump always vanishes on the specially designed adaptive mesh. Consequently, the discretization of the underlying model equation is further simplified while maintaining a second-order convergence rate in space. Moreover, the derived discretization is reinterpreted from a novel perspective, which shows some inspiration to extend the proposed method to a multi-dimensional case.
- Thirdly, with the help of exact tracing of each heat source by a fixed mesh point, the blow-up phenomenon that occurs at multiple locations simultaneously has been successfully observed by the present numerical experiment of symmetrically moving sources.
- Finally, various numerical experiments are carried out to investigate the solution behavior in terms of the motions of heat sources, such as the types of motion and the distance between the sources. The blow-up phenomena in terms of the distance of two moving sources are numerically discussed. It is found as expected that, if the sources are moving with high speeds, there may exist a critical distance, such that blow-up must occur if the distance of the sources is less than the critical distance, while blow-up is avoided when the distance is larger than this critical distance.
2. Model Equation
3. Numerical Method
3.1. Domain-Decomposed Adaptive Mesh Generation
3.2. Discretization on the Moving Mesh
3.3. Numerical Algorithm
4. Numerical Examples and Discussion
5. Conclusions
Funding
Acknowledgments
Conflicts of Interest
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N, L | Ratio | Ratio | Ratio | Ratio | ||||
---|---|---|---|---|---|---|---|---|
40, 40 | 8.2720 × | - | 1.0931 × | - | 1.3721 × | - | 6.5460 × | - |
80, 160 | 2.0540 × | 4.0274 | 2.6996 × | 4.0491 | 3.3908 × | 4.0465 | 6.0108 × | 1.0890 |
160, 640 | 5.1038 × | 4.0245 | 6.6945 × | 4.0326 | 8.4144 × | 4.0297 | 1.1557 × | 5.2009 |
320, 2560 | 1.2732 × | 4.0085 | 1.6687 × | 4.0117 | 2.0981 × | 4.0104 | 2.9347 × | 3.9381 |
640, 10,240 | 3.1807 × | 4.0029 | 4.1678 × | 4.0038 | 5.2408 × | 4.0035 | 7.3175 × | 4.0106 |
1280, 40,960 | 7.9500 × | 4.0010 | 1.0416 × | 4.0013 | 1.3098 × | 4.0011 | 1.8237 × | 4.0124 |
2560, 163,840 | 1.9874 × | 4.0003 | 2.6038 × | 4.0003 | 3.2743 × | 4.0003 | 4.5462 × | 4.0115 |
2.03881627 | 1.14555847 | 0.48168300 | ||||||
5 | 3.10495654 | 3.49923032 | 0.61947967 | |||||
10 | 5.21961249 | 10.60946467 | 0.88168884 | |||||
25 | 11.64283736 | 16.52257933 | - | 0.78709834 | ||||
50 | 22.56509042 | - | 1.40807506 | - | 0.76986615 | |||
100 | 44.58805080 | - | 1.26522313 | - | 0.71490192 |
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Hu, Z. Numerical Investigation of Heat Conduction with Multiple Moving Heat Sources. Symmetry 2018, 10, 673. https://doi.org/10.3390/sym10120673
Hu Z. Numerical Investigation of Heat Conduction with Multiple Moving Heat Sources. Symmetry. 2018; 10(12):673. https://doi.org/10.3390/sym10120673
Chicago/Turabian StyleHu, Zhicheng. 2018. "Numerical Investigation of Heat Conduction with Multiple Moving Heat Sources" Symmetry 10, no. 12: 673. https://doi.org/10.3390/sym10120673
APA StyleHu, Z. (2018). Numerical Investigation of Heat Conduction with Multiple Moving Heat Sources. Symmetry, 10(12), 673. https://doi.org/10.3390/sym10120673