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]**Jim Douglas Jr.,*On the numerical integration of ∂²𝑢/∂𝑥²+∂²𝑢/∂𝑦²=∂𝑢/∂𝑡 by implicit methods*, J. Soc. Indust. Appl. Math.**3**(1955), 42–65. MR**71875****[3]**Jim Douglas Jr.,*On the numerical integration of quasilinear parabolic differential equations*, Pacific J. Math.**6**(1956), 35–42. MR**79196****[4]**Jim Douglas Jr.,*The solution of the diffusion equation by a high order correct difference equation*, J. Math. and Phys.**35**(1956), 145–151. MR**90875**, https://doi.org/10.1002/sapm1956351145**[5]**Jim Douglas Jr.,*On the relation between stability and convergence in the numerical solution of linear parabolic and hyperbolic differential equations*, J. Soc. Indust. Appl. Math.**4**(1956), 20–37. MR**80368****[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]**Jim 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.**82**(1956), 421–439. MR**84194**, https://doi.org/10.1090/S0002-9947-1956-0084194-4**[9]**Paul R. Halmos,*Finite Dimensional Vector Spaces*, Annals of Mathematics Studies, no. 7, Princeton University Press, Princeton, N.J., 1942. MR**0006591****[10]**M. R. Hestenes and W. Karush,*Solutions of 𝐴𝑥=𝜆𝐵𝑥*, J. Research Nat. Bur. Standards**47**(1951), 471–478. MR**0049852****[11]**Fritz John,*On integration of parabolic equations by difference methods. I. Linear and quasi-linear equations for the infinite interval*, Comm. Pure Appl. Math.**5**(1952), 155–211. MR**47885**, https://doi.org/10.1002/cpa.3160050203**[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.**9**(1956), 267–293. MR**79204**, https://doi.org/10.1002/cpa.3160090206**[14]**Solomon Lefschetz,*Introduction to Topology*, Princeton Mathematical Series, vol. 11, Princeton University Press, Princeton, N. J., 1949. MR**0031708****[15]**William Edmund Milne,*Numerical solution of differential equations*, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR**0068321****[16]**George G. O’Brien, Morton A. Hyman, and Sidney Kaplan,*A study of the numerical solution of partial differential equations*, J. Math. Physics**29**(1951), 223–251. MR**0040805****[17]**D. W. Peaceman and H. H. Rachford Jr.,*The numerical solution of parabolic and elliptic differential equations*, J. Soc. Indust. Appl. Math.**3**(1955), 28–41. MR**71874****[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