The application of stability analysis in the numerical solution of quasi-linear parabolic differential equations

Author:
Jim Douglas

Journal:
Trans. Amer. Math. Soc. **89** (1958), 484-518

MSC:
Primary 65.68; Secondary 35.65

DOI:
https://doi.org/10.1090/S0002-9947-1958-0131673-9

MathSciNet review:
0131673

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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. 42-65. MR**0071875 (17:196e)****[3]**-,*On the numerical integration of quasi-linear parabolic differential equations*, Pacific J. Math. vol. 6 (1956) pp. 35-42. 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. 145-152. 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. 20-37. 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 unsteady-state gas flow within a square drainage area*, Trans. Amer. Inst. Mining, Metallurgical, and Petroleum Engineers, vol. 204 (1955) pp. 190-195.**[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. 421-439. 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. 471-478. MR**0049852 (14:236b)****[11]**F. John,*On integration of parabolic differential equations by difference methods*, Comm. Pure Appl. Math. vol. 5 (1952) pp. 155-211. MR**0047885 (13:947b)****[12]**M. L. Juncosa and D. Young,*On the Crank-Nicolson procedure for parabolic differential equations*, Bull. Amer. Math. Soc. Abstract 60-2-240.**[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. 267-293. 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. 223-251. 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. 28-41. MR**0071874 (17:196d)****[18]**A. Zygmund,*Trigonometrical series*, Warsaw-Lwow, 1935.

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DOI:
https://doi.org/10.1090/S0002-9947-1958-0131673-9

Article copyright:
© Copyright 1958
American Mathematical Society