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The new alternating direction implicit difference methods for solving three-dimensional parabolic equations. (English) Zbl 1185.65151
Summary: A new alternating direction implicit (ADI) scheme for solving three-dimensional parabolic equations with nonhomogeneous boundary conditions is presented. The scheme is also extended to high-order compact difference scheme. Both of them have the advantages of unconditional stability and being convenient to compute the boundary values of the intermediates. Besides this, the compact scheme has high-order accuracy and uses less computational time. Numerical examples are presented and the results are very satisfactory.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] Peaceman, D.W.; Rachford, H., The numerical solution of parabolic and elliptic differential equations, J. soc. indust. appl. math., 3, 28-41, (1959) · Zbl 0067.35801
[2] Douglas, J.; Peaceman, D.W., Numerical solution for two-dimensional heat flow problems, Am. inst. chem. eng. J., 1, 505-512, (1955)
[3] D’yakonov, E., Difference schemes with splitting operators for multidimensional unsteady problems (English translation), USSR comput. math., 3, 581-607, (1963)
[4] Douglas Jr, J.; Kim, S., Improved accuracy for locally one-dimensional methods for parabolic equations, Math. mod. meth. appl. sci., 11, 9, 1563-1579, (2001) · Zbl 1012.65095
[5] Dehghan, M., A new ADI technique for two-dimensional parabolic equation with an integral condition, Comput. math. appl., 43, 1477-1488, (2002) · Zbl 1001.65094
[6] Wang, C.; Wang, T., Extended locally one-dimensional finite difference and finite element schemes for nonhomogeneous parabolic differential equations with nonhomogeneous boundary conditions, Numer. math. J. Chinese univ., 28, 2, 138-150, (2006) · Zbl 1115.65357
[7] Dehghan, M., Numerical solution of the three-dimensional parabolic equation with an integral condition, Numer. meth. part DE, 18, 193-202, (2002) · Zbl 0996.65079
[8] Dehghan, M., Locally explicit schemes for three-dimensional diffusion with a nonlocal boundary specification, Appl. math. comput., 138, 489-501, (2003) · Zbl 1027.65112
[9] Dehghan, M., Fractional step methods for parabolic equations with a non-standard condition, Appl. math. comput., 142, 177-187, (2003) · Zbl 1022.65088
[10] Qin, J.; Wang, X.; Wang, T., A new alternating direction implicit difference method for three-dimensional parabolic differential equations with homogeneous boundary conditions, Chinese J. hydrodynam., 23, 393-403, (2008)
[11] Karaa, S., An accurate LOD scheme for two-dimensional parabolic problems, Appl. math. comput., 170, 886-894, (2005) · Zbl 1103.65094
[12] Dai, W.; Nassar, R., Compact ADI method for solving parabolic differential equations, Numer. meth. part DE, 18, 2, 129-142, (2002) · Zbl 1004.65086
[13] Karaa, S., A high-order compact ADI method for solving three-dimensional unsteady convection – diffusion problems, Numer. meth. part DE, 22, 983-993, (2006) · Zbl 1099.65074
[14] Zhang, J., Multigrid method and fourth-order compact scheme for 2D Poisson equation with uneaqual mesh-size discretiation, J. comput. phys., 179, 170-179, (2002) · Zbl 1005.65137
[15] Li, J.; Chen, Y.; Liu, G., High-order compact ADI methods for parabolic equations, Comput. math. appl., 52, 1343-1356, (2006) · Zbl 1121.65092
[16] Karaa, S.; Zhang, J., High order ADI method for solving unsteady convection – diffusion problems, J. comput. phys., 198, 1-9, (2004) · Zbl 1053.65067
[17] Tian, Z.F.; Ge, Y.B., A fourth-order compact ADI method for solving two-dimensional unsteady convection – diffusion problems, J. comput. appl. math., 198, 268-286, (2007) · Zbl 1104.65086
[18] Kalita, J.C.; Dalal, D.C.; Dass, A.K., A class of higher order compact scheme for the unsteady two-dimensional convection – diffusion equation with variable convection coefficients, Int. J. numer. meth. fluids, 38, 1111-1131, (2002) · Zbl 1094.76546
[19] Lele, S.K., Compact finite difference schemes with spectral-like solution, J. comput. phys., 103, 16-42, (1992) · Zbl 0759.65006
[20] Yu, D.; Tang, H., Numerical methods for differential equations (in Chinese), (2003), China Science Press
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