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Volumn 26, Issue 1, 2010, Pages 37-60

Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations

Author keywords

ADI scheme; Asymptotic expansion; Compact ADI scheme; Discrete energy method; Parabolic equation; Richardson extrapolation

Indexed keywords


EID: 73449143894     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20414     Document Type: Article
Times cited : (172)

References (29)
  • 1
    • 0042939980 scopus 로고    scopus 로고
    • An ADI method for hysteretic reaction‐diffusion systems
    • 1 C. Chiu and N. Walkington, An ADI method for hysteretic reaction‐diffusion systems, SIAM J Numer Anal 34 ( 1997), 1185–1206.
    • (1997) SIAM J Numer Anal , vol.34 , pp. 1185-1206
    • Chiu, C.1    Walkington, N.2
  • 2
    • 0008820821 scopus 로고    scopus 로고
    • A second‐order ADI scheme for three‐dimensional parabolic differential equations
    • 2 W. Z. Dai, A second‐order ADI scheme for three‐dimensional parabolic differential equations, Numer Methods Partial Differential Eq 14 ( 1998), 159–168.
    • (1998) Numer Methods Partial Differential Eq , vol.14 , pp. 159-168
    • Dai, W.Z.1
  • 3
    • 34250949893 scopus 로고
    • Alternating direction method for three space variables
    • 3 J. Jr. Douglas, Alternating direction method for three space variables, Numer Math 4 ( 1961), 41–63.
    • (1961) Numer Math , vol.4 , pp. 41-63
    • Douglas, J.1
  • 4
    • 0001742634 scopus 로고
    • A general formulation of alternating direction method. I. Parabolic and hyperbolic problem
    • 4 J. Jr. Douglas and J. Gunn, A general formulation of alternating direction method. I. Parabolic and hyperbolic problem, Numer Math 6 ( 1964), 428–453.
    • (1964) Numer Math , vol.6 , pp. 428-453
    • Douglas, J.1    Gunn, J.2
  • 5
    • 0002058827 scopus 로고
    • The numerical solution of parabolic and elliptic differential equations
    • 5 D. W. Peacemann and H. H. Jr. Rachford, The numerical solution of parabolic and elliptic differential equations, J Soc Indust Appl Math 3 ( 1955), 28–41.
    • (1955) J Soc Indust Appl Math , vol.3 , pp. 28-41
    • Peacemann, D.W.1    Rachford, H.H.2
  • 6
    • 84924514108 scopus 로고    scopus 로고
    • Numerical solution of partial differential equations
    • 6 K. W. Morton and D. F. Mayers, Numerical solution of partial differential equations, Cambridge University Press, Cambridge, 2005.
    • (2005)
    • Morton, K.W.1    Mayers, D.F.2
  • 7
    • 3042753774 scopus 로고
    • Difference schemes of second‐order accuracy with a splitting operator for parabolic equations without mixed partial derivatives
    • 7 E. G. D'yakonov, Difference schemes of second‐order accuracy with a splitting operator for parabolic equations without mixed partial derivatives, Zh Vychisl Mat Mat Fiz 4 ( 1964), 935–941.
    • (1964) Zh Vychisl Mat Mat Fiz , vol.4 , pp. 935-941
    • D'yakonov, E.G.1
  • 8
    • 0003985596 scopus 로고
    • The method of fractional steps
    • 8 N. N. Yanenko, The method of fractional steps, Springer‐Verlag, Berlin, 1971.
    • (1971)
    • Yanenko, N.N.1
  • 9
    • 0036495593 scopus 로고    scopus 로고
    • Compact ADI method for solving parabolic differential equations
    • 9 W. Z. Dai, Compact ADI method for solving parabolic differential equations, Numer Methods Partial Differential Eq 18 ( 2002), 129–142.
    • (2002) Numer Methods Partial Differential Eq , vol.18 , pp. 129-142
    • Dai, W.Z.1
  • 10
    • 0001642489 scopus 로고
    • A new computational procedure for A.D.I. methods
    • 10 G. Fairweather, A new computational procedure for A.D.I. methods, SIAM J Numer Anal 4 ( 1967), 163–170.
    • (1967) SIAM J Numer Anal , vol.4 , pp. 163-170
    • Fairweather, G.1
  • 11
    • 3042771625 scopus 로고    scopus 로고
    • High‐order ADI method for solving unsteady convection‐diffusion problems
    • 11 S. Karaa and J. Zhang, High‐order ADI method for solving unsteady convection‐diffusion problems, J Comput Phys 198 ( 2004), 1–9.
    • (2004) J Comput Phys , vol.198 , pp. 1-9
    • Karaa, S.1    Zhang, J.2
  • 12
    • 33745799513 scopus 로고    scopus 로고
    • A high‐order compact ADI method for solving three‐dimensional unsteady convection‐diffusion problems
    • 12 S. Karaa, A high‐order compact ADI method for solving three‐dimensional unsteady convection‐diffusion problems, Numer Methods Partial Diff Eq 22 ( 2006), 983–993.
    • (2006) Numer Methods Partial Diff Eq , vol.22 , pp. 983-993
    • Karaa, S.1
  • 13
    • 85120595327 scopus 로고    scopus 로고
    • A compact alternate direction implicit difference method for the reaction diffusion equations with variable coefficients
    • 13 X. L. Li and Z. Z. Sun, A compact alternate direction implicit difference method for the reaction diffusion equations with variable coefficients, Numer Math: J Chinese Univ 28 ( 2006), 83–95, (in Chinese).
    • (2006) Numer Math: J Chinese Univ , vol.28 , pp. 83-95
    • Li, X.L.1    Sun, Z.Z.2
  • 14
    • 0003095767 scopus 로고
    • Improved forms of the alternating direction methods of Douglas, Peaceman, and Rachford for soling parabolic and elliptic equations
    • 14 A. R. Mitchell and G. Fairweather, Improved forms of the alternating direction methods of Douglas, Peaceman, and Rachford for soling parabolic and elliptic equations, Numer Math 6 ( 1964), 285–292.
    • (1964) Numer Math , vol.6 , pp. 285-292
    • Mitchell, A.R.1    Fairweather, G.2
  • 15
    • 54549108683 scopus 로고    scopus 로고
    • A compact alternating direction implicit difference method for reaction diffusion equations
    • 15 Z. Z. Sun and X. L. Li, A compact alternating direction implicit difference method for reaction diffusion equations, Math Numer Sin 27 ( 2005), 209–224, (in Chinese).
    • (2005) Math Numer Sin , vol.27 , pp. 209-224
    • Sun, Z.Z.1    Li, X.L.2
  • 16
    • 84953600891 scopus 로고
    • A paractial method for numerical evaluation of solutions of partial differential equations of the heat‐conduction type
    • 16 J. Crank and P. Nicolson, A paractial method for numerical evaluation of solutions of partial differential equations of the heat‐conduction type, Proc Camb Philos Soc 43 ( 1947), 50–67.
    • (1947) Proc Camb Philos Soc , vol.43 , pp. 50-67
    • Crank, J.1    Nicolson, P.2
  • 17
    • 0037197866 scopus 로고    scopus 로고
    • A class of higer order compact schemes for the unsteady two‐dimensional convection‐diffusion equation with variable convection coefficients
    • 17 J. C. Kalita and A. K. Dass, A class of higer order compact schemes for the unsteady two‐dimensional convection‐diffusion equation with variable convection coefficients, Int J Numer Methods Fluids 38 ( 2002), 1111–1131.
    • (2002) Int J Numer Methods Fluids , vol.38 , pp. 1111-1131
    • Kalita, J.C.1    Dass, A.K.2
  • 18
    • 0029106259 scopus 로고
    • A compact fouth‐order finite difference scheme for the steady incompressible Navier‐Stokes equations
    • 18 M. Li, T. Tang, and B. Fornberg, A compact fouth‐order finite difference scheme for the steady incompressible Navier‐Stokes equations, Int J Num Methods Fluids 20 ( 1995), 1137–1151.
    • (1995) Int J Num Methods Fluids , vol.20 , pp. 1137-1151
    • Li, M.1    Tang, T.2    Fornberg, B.3
  • 19
    • 0036569404 scopus 로고    scopus 로고
    • An efficient high‐order algorithm for solving systems of reaction‐diffusion equations
    • 19 W. Y. Liao and A. Q. M. Khaliq, An efficient high‐order algorithm for solving systems of reaction‐diffusion equations, Numer Methods Partial Differential Eq 18 ( 2006), 340–354.
    • (2006) Numer Methods Partial Differential Eq , vol.18 , pp. 340-354
    • Liao, W.Y.1    Khaliq, A.Q.M.2
  • 20
    • 0035498905 scopus 로고    scopus 로고
    • An unconditionally stable and O(τ2 + h4) order L∞ convergent difference scheme for parabolic equations with variable coefficients
    • 20 Z. Z. Sun, An unconditionally stable and O (τ 2 + h 4) order L ∞ convergent difference scheme for parabolic equations with variable coefficients, Numer Methods Partial Differential Eq 17 ( 2001), 619–631.
    • (2001) Numer Methods Partial Differential Eq , vol.17 , pp. 619-631
    • Sun, Z.Z.1
  • 21
    • 34547813971 scopus 로고    scopus 로고
    • Numerical methods of partial differential equations
    • 21 Z. Z. Sun, Numerical methods of partial differential equations, Science Press, Beijing, (inChinese), 2005.
    • (2005)
    • Sun, Z.Z.1
  • 22
    • 0003796630 scopus 로고
    • Sobolev spaces
    • 22 R. A. Adams, Sobolev spaces, Academic Press, New York, 1975.
    • (1975)
    • Adams, R.A.1
  • 23
    • 0003994311 scopus 로고
    • Difference methods and their extrapolations
    • 23 G. I. Marchuk and V. V. Shaidurov, Difference methods and their extrapolations, Springer‐Verlag, New York, 1983.
    • (1983)
    • Marchuk, G.I.1    Shaidurov, V.V.2
  • 24
    • 0003881703 scopus 로고
    • Difference methods for elliptic equations
    • 24 A. A. Samarskii and V. B. Andreev, Difference methods for elliptic equations, Nauka, Moscow, 1976.
    • (1976)
    • Samarskii, A.A.1    Andreev, V.B.2
  • 25
    • 0003947270 scopus 로고
    • Applications of discrete functional analysis to the finite difference method
    • 25 Y. L. Zhou, Applications of discrete functional analysis to the finite difference method, International Academic Publishers, Beijing, 1990.
    • (1990)
    • Zhou, Y.L.1
  • 26
    • 0037726571 scopus 로고    scopus 로고
    • Some discrete interpolation inequalities with several indices
    • 26 Y. L. Zhou and G. W. Yuan, Some discrete interpolation inequalities with several indices, Beijing Math 2 ( 1996), 94–108.
    • (1996) Beijing Math , vol.2 , pp. 94-108
    • Zhou, Y.L.1    Yuan, G.W.2
  • 27
    • 0003430389 scopus 로고    scopus 로고
    • Numerical approximation of partial differential equations
    • 27 A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, Springer‐Verlag, New York, 1997.
    • (1997)
    • Quarteroni, A.1    Valli, A.2
  • 28
    • 0003423678 scopus 로고
    • Numerical partial differential equations: finite difference methods
    • 28 J. W. Thomas, Numerical partial differential equations: finite difference methods, Springer‐Verlag, New York, 1995.
    • (1995)
    • Thomas, J.W.1
  • 29
    • 9144220381 scopus 로고
    • Compact finite difference schemes with spectral‐like resolution
    • 29 S. K. Lele, Compact finite difference schemes with spectral‐like resolution, J Comput Phys 103 ( 1992), 16–42.
    • (1992) J Comput Phys , vol.103 , pp. 16-42
    • Lele, S.K.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.