zbMATH — the first resource for mathematics

High-order compact ADI methods for parabolic equations. (English) Zbl 1121.65092
Summary: We develop a sixth-order compact scheme coupled with alternating direction implicit (ADI) methods and apply it to parabolic equations in both 2-D and 3-D. Unconditional stability is proved for linear diffusion problems with periodic boundary conditions. Numerical examples supporting our theoretical analysis are provided.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65F05 Direct numerical methods for linear systems and matrix inversion
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI
[1] Lele, S.K., Compact finite difference schemes with spectral-like solution, J. comp. phys., 103, 16-42, (1992) · Zbl 0759.65006
[2] Hixon, R.; Turkel, E., Compact implicit maccormack-type schemes with high accuracy, J. comp. phys., 158, 51-70, (2000) · Zbl 0958.76059
[3] Hu, F.Q.; Hussaini, M.Y.; Manthey, J.L., Low-dissipation and low- dispersion Runge-Kutta schemes for computational acoustics, J. comp. phys., 124, 177-191, (1996) · Zbl 0849.76046
[4] Tarn, C.K.W.; Webb, J.C., Dispersion-relation-preserving finite difference schemes for computational acoustics, J. comp. phys., 107, 262-281, (1993) · Zbl 0790.76057
[5] Shang, J.S., High-order compact-difference schemes for time- dependent Maxwell equations, J. comp. phys., 153, 312-333, (1999) · Zbl 0956.78018
[6] Li, J.; Chen, Y., High-order compact schemes for dispersive media, Electronics letters, 40, 853-855, (2004)
[7] Navon, I.M.; Riphagen, H.A., An implicit compact fourth order algorithm for solving the shallow-water equations in conservation-law form, Monthly weather review, 107, 1107-1127, (1979) · Zbl 0434.65079
[8] Navon, I.M.; Riphagen, H.A., SHALL4—an implicit compact fourth-order Fortran program for solving the shallow-water equations in conservation-law form, Computers & geosciences, 12, 129-150, (1986)
[9] Chu, P.C.; Fan, C., A three-point combined compact difference scheme, J. comp. phys., 140, 370-399, (1998) · Zbl 0923.65071
[10] Cockburn, B.; Shu, C.-W., Nonlinearly stable compact schemes for shock calculations, SIAM J. numer. anal., 31, 607-627, (1994) · Zbl 0805.65085
[11] Visbal, M.R.; Gaitonde, D.V., High-order-accurate methods for complex unsteady subsonic flows, Aiaa j., 37, 1231-1239, (1999)
[12] Abarbanel, S.; Ditkowski, A.; Gustafsson, B., On error bounds of finite difference approximations to partial differential equations—temporal behavior and rate of convergence, J. scientific computing, 15, 79-116, (2000) · Zbl 0984.65095
[13] Carpenter, M.H.; Gottlieb, D.; Abarbanel, S., Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact system, J. comp. phys., 111, 220-236, (1994) · Zbl 0832.65098
[14] Carpenter, M.H.; Gottlieb, D.; Abarbanel, S., Stable and accurate boundary treatments for compact high order finite difference schemes, Appl. numer. math., 12, 55-87, (1993) · Zbl 0778.65057
[15] Adam, Y., Highly accurate compact implicit methods and boundary conditions, J. comp. phys., 24, 10-22, (1977) · Zbl 0357.65074
[16] Spotz, W.F.; Carey, G.F., Extension of high order compact schemes to time dependent problems, Numer. methods for pdes, 17, 657-672, (2001) · Zbl 0998.65101
[17] Dai, W.; Nassar, R., Compact ADI method for solving parabolic differential equations, Numer. methods for pdes, 18, 129-142, (2002) · Zbl 1004.65086
[18] Liao, W.; Zhu, J.; Khaliq, A.Q.M., An efficient high-order algorithm for solving systems of reaction-diffusion equations, Numer. methods for pdes, 18, 340-354, (2002) · Zbl 0997.65105
[19] Karaa, S.; Zhang, J., High order ADI method for solving unsteady convection-diffusion problems, J. comp. phys., 198, 1-9, (2004) · Zbl 1053.65067
[20] Mattsson, K.; Nordstrom, J., Summation by parts operators for finite difference approximations of second derivatives, J. comp. phys., 199, 503-540, (2004) · Zbl 1071.65025
[21] Li, J.; Visbal, M.R., High-order compact schemes for nonlinear dispersive waves, J. scientific computing, 26, 1-23, (2006) · Zbl 1089.76043
[22] Li, J., High-order finite difference schemes for differential equations containing higher derivatives, Appl. math. comp., 171, 1157-1176, (2005) · Zbl 1090.65101
[23] D.V. Gaitonde and M.R. Visbal, High-order schemes for Navier-Stokes equations: Algorithms and implementation into FDL3DI, Technical Report AFRL-VA-WP-TR-1998-3060, Air Force Research Laboratory, Wright-Patterson AFB, OH, (1998).
[24] Peaceman, D.W.; Rachford, H.H., The numerical solution of parabolic and elliptic differential equations, J. soc. indust. appl. math., 3, 28-41, (1955) · Zbl 0067.35801
[25] Douglas, J., Alternating direction methods for three space variables, Numer. math., 4, 41-63, (1962) · Zbl 0104.35001
[26] Douglas, J.; Gunn, J.E., A general formulation of alternating direction methods—part I. parabolic and hyperbolic problems, Numer. math., 6, 428-453, (1964) · Zbl 0141.33103
[27] Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time dependent problems and difference methods, (1995), Wiley New York
[28] Lapidus, L.; Pinder, G., Numerical solution of partial differential equations in science and engineering, (1982), Wiley New York · Zbl 0584.65056
[29] Li, J.; Hon, Y.C.; Chen, C.S., Numerical comparisons of two meshless methods using radial basis functions, Engineering analysis with boundary elements, 26, 205-225, (2002) · Zbl 1003.65132
[30] Fairweather, G.; Mitchell, A.R., A new computational procedure for A.D.I, methods, SIAM J. numer. anal., 4, 163-170, (1967) · Zbl 0252.65072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.