The application of stability analysis in the numerical solution of quasilinear parabolic differential equations
Author:
Jim Douglas
Journal:
Trans. Amer. Math. Soc. 89 (1958), 484518
MSC:
Primary 65.68; Secondary 35.65
MathSciNet review:
0131673
Fulltext PDF Free Access
References 
Similar Articles 
Additional Information
 [1]
R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, New York, 1953.
 [2]
J. Douglas, Jr., On the numerical integration of by implicit methods, J. Soc. Indust. Appl. Math. vol. 3 (1955) pp. 4265. MR 0071875 (17:196e)
 [3]
, On the numerical integration of quasilinear parabolic differential equations, Pacific J. Math. vol. 6 (1956) pp. 3542. MR 0079196 (18:46g)
 [4]
, The solution of the diffusion equation by a high order correct difference equation, J. Math. Phys. vol. 35 (1956) pp. 145152. MR 0090875 (19:884f)
 [5]
, On the relation between stability and convergence in the numerical solution of linear parabolic and hyperbolic differential equations, J. Soc. Indust. Appl. Math. vol. 4 (1956) pp. 2037. MR 0080368 (18:236d)
 [6]
, A note on the numerical solution of parabolic differential equations, to appear.
 [7]
J. Douglas, Jr., D. W. Peaceman, and H. H. Rachford, Jr., Calculation of unsteadystate gas flow within a square drainage area, Trans. Amer. Inst. Mining, Metallurgical, and Petroleum Engineers, vol. 204 (1955) pp. 190195.
 [8]
J. Douglas, Jr. and H. H. Rachford, Jr., On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc. vol. 82 (1956) pp. 421439. MR 0084194 (18:827f)
 [9]
P. R. Halmos, Finite dimensional vector spaces, Princeton, 1942. MR 0006591 (4:11a)
 [10]
M. R. Hestenes and W. Karush, Solutions of , J. Res. Nat. Bur. Standards vol. 47 (1951) pp. 471478. MR 0049852 (14:236b)
 [11]
F. John, On integration of parabolic differential equations by difference methods, Comm. Pure Appl. Math. vol. 5 (1952) pp. 155211. MR 0047885 (13:947b)
 [12]
M. L. Juncosa and D. Young, On the CrankNicolson procedure for parabolic differential equations, Bull. Amer. Math. Soc. Abstract 602240.
 [13]
P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. vol. 9 (1956) pp. 267293. MR 0079204 (18:48c)
 [14]
S. Lefschetz, Introduction to topology, Princeton, 1949. MR 0031708 (11:193e)
 [15]
W. E. Milne, Numerical solution of differential equations, New York, 1953. MR 0068321 (16:864c)
 [16]
G. G. O'Brien, M. A. Hyman, and S. Kaplan, A study of the numerical solution of partial differential equations, J. Math. Phys. vol. 29 (1951) pp. 223251. MR 0040805 (12:751e)
 [17]
D. W. Peaceman and H. H. Rachford, Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. vol. 3 (1955) pp. 2841. MR 0071874 (17:196d)
 [18]
A. Zygmund, Trigonometrical series, WarsawLwow, 1935.
 [1]
 R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, New York, 1953.
 [2]
 J. Douglas, Jr., On the numerical integration of by implicit methods, J. Soc. Indust. Appl. Math. vol. 3 (1955) pp. 4265. MR 0071875 (17:196e)
 [3]
 , On the numerical integration of quasilinear parabolic differential equations, Pacific J. Math. vol. 6 (1956) pp. 3542. MR 0079196 (18:46g)
 [4]
 , The solution of the diffusion equation by a high order correct difference equation, J. Math. Phys. vol. 35 (1956) pp. 145152. MR 0090875 (19:884f)
 [5]
 , On the relation between stability and convergence in the numerical solution of linear parabolic and hyperbolic differential equations, J. Soc. Indust. Appl. Math. vol. 4 (1956) pp. 2037. MR 0080368 (18:236d)
 [6]
 , A note on the numerical solution of parabolic differential equations, to appear.
 [7]
 J. Douglas, Jr., D. W. Peaceman, and H. H. Rachford, Jr., Calculation of unsteadystate gas flow within a square drainage area, Trans. Amer. Inst. Mining, Metallurgical, and Petroleum Engineers, vol. 204 (1955) pp. 190195.
 [8]
 J. Douglas, Jr. and H. H. Rachford, Jr., On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc. vol. 82 (1956) pp. 421439. MR 0084194 (18:827f)
 [9]
 P. R. Halmos, Finite dimensional vector spaces, Princeton, 1942. MR 0006591 (4:11a)
 [10]
 M. R. Hestenes and W. Karush, Solutions of , J. Res. Nat. Bur. Standards vol. 47 (1951) pp. 471478. MR 0049852 (14:236b)
 [11]
 F. John, On integration of parabolic differential equations by difference methods, Comm. Pure Appl. Math. vol. 5 (1952) pp. 155211. MR 0047885 (13:947b)
 [12]
 M. L. Juncosa and D. Young, On the CrankNicolson procedure for parabolic differential equations, Bull. Amer. Math. Soc. Abstract 602240.
 [13]
 P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. vol. 9 (1956) pp. 267293. MR 0079204 (18:48c)
 [14]
 S. Lefschetz, Introduction to topology, Princeton, 1949. MR 0031708 (11:193e)
 [15]
 W. E. Milne, Numerical solution of differential equations, New York, 1953. MR 0068321 (16:864c)
 [16]
 G. G. O'Brien, M. A. Hyman, and S. Kaplan, A study of the numerical solution of partial differential equations, J. Math. Phys. vol. 29 (1951) pp. 223251. MR 0040805 (12:751e)
 [17]
 D. W. Peaceman and H. H. Rachford, Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. vol. 3 (1955) pp. 2841. MR 0071874 (17:196d)
 [18]
 A. Zygmund, Trigonometrical series, WarsawLwow, 1935.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947195801316739
PII:
S 00029947(1958)01316739
Article copyright:
© Copyright 1958
American Mathematical Society
