# Conservative domain decomposition schemes for solving two-dimensional heat equations

- 63 Downloads

## Abstract

In this paper, by combining the operator splitting technique, a new mass-conserved domain decomposition method for two-dimensional heat equations is proposed. Along the each direction, the interface fluxes are first calculated from the explicit fluxes, then the sub-domain’s interior solutions are paralelly computed by the C–N implicit scheme. The scheme is stable under the condition \(r\le 2(\sqrt{6}-2)\) and the corresponding convergence order of the scheme are given in \(L^2\)-norm. Numerical results confirm the theoretical results.

## Keywords

Mass-conserved Domain decompositions Interface fluxes \(L^2\)-norm## Mathematics Subject Classification

65M06 65M12 65M55 76S05## Notes

### Acknowledgements

This work was supported by Natural Science Foundation of China (Grant Nos. 6170325, 61503227), and Natural Science Foundation of Shandong Government (Grant Nos. ZR2017BA029, ZR2017BF002), Shandong Agricultural University (Grant No. xxxy201704), and National natural science foundation funding project application for key subject.

## References

- Dawson C, Du Q, Dupont T (1991) A finite difference domain decomposition algorithm for numerical solution of heat equations. Math Comput 57:63–71MathSciNetCrossRefGoogle Scholar
- Dawson C, Dupont T (1994) Explicit/implicit conservative domain decomposition procedures for parabolic problems based on block centered finite differences. SIAM J Numer Anal 31:1045–1061MathSciNetCrossRefGoogle Scholar
- Du C, Liang D (2010) An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media. J Comput Phys 229:4501–4521MathSciNetCrossRefGoogle Scholar
- Du Q, Mu M, Wu Z (2001) Efficient parallel algorithms for parbolic problems. SIAM J Numer Anal 39:1469–1487MathSciNetCrossRefGoogle Scholar
- Jia D, Sheng Z, Yuan G (2018) A conservative parallel difference method for 2-dimension diffusion equation. Appl Math Lett 78:72–78MathSciNetCrossRefGoogle Scholar
- Liang D, Du C (2014) The efficient S-DDM scheme and its analysis for solving parabolic equations. J Comput Phys 272:46–69MathSciNetCrossRefGoogle Scholar
- Shi H, Liao H (2006) Unconditional stability of corrected explicit/implicit domain decomposition algorithms for parallel approximation of heat equations. SIAM J Numer Anal 44:1584–1611MathSciNetCrossRefGoogle Scholar
- Sheng Z, Yuan G, Hang X (2007) Unconditional stability of parallel difference schemes with second order accuracy for parabolic equation. Appl Math Comput 184:1015–1031MathSciNetzbMATHGoogle Scholar
- Yuan G, Yao Y, Yin L (2011) A Conservative domain decomposition produce for nonlinear diffusion problems on arbitrary quadrilateral grids. SIAM J Sci Comput 33:1352–1368MathSciNetCrossRefGoogle Scholar
- Yu Y, Yao Y, Yuan G, Chen X (2016) A conservative parallel iteration scheme for nonlinear diffusion equations on unstructured meshes. Commun Comput Phys 20:1405–1423MathSciNetCrossRefGoogle Scholar
- Zhou Z, Liang D (2016) The mass-preserving S-DDM scheme for two-dimensional parabolic equations. Commun Comput Phys 19:411–441MathSciNetCrossRefGoogle Scholar
- Zhou Z, Liang D, Wong Y (2018) The new mass-conserving S-DDM scheme for two-dimensional parabolic equations with variable coefficients. Appl Math Comput 338:882–902MathSciNetGoogle Scholar
- Zhou Z, Liang D (2017) The mass-preserving and modified-upwind splitting DDM scheme for time-dependent convection-diffusion equations. J Comput Appl Math 317:247–273MathSciNetCrossRefGoogle Scholar
- Zhou Z, Liang D (2017) A time second-order mass-conserved implicit-explicit domain decomposition scheme for solving the diffusion equations. Adv Appl Math Mech 9:1–23MathSciNetCrossRefGoogle Scholar
- Zhou Z, Liang D (2018) Mass-preserving time second-order explicit-implicit domain decomposition schemes for solving parabolic equations with variable coefficients. Comput Appl Math 37:4423–4442MathSciNetCrossRefGoogle Scholar
- Zhuang Y, Sun X (2002) Stabilitized explicit-implicit domain decomposition methods for the numerical solution of parabolic equations. SIAM J Sci Comput 24:335–358MathSciNetCrossRefGoogle Scholar