A posteriori bounds in the numerical solution of mildly nonlinear parabolic equations
Author:
Alfred Carasso
Journal:
Math. Comp. 24 (1970), 785792
MSC:
Primary 65.68
MathSciNet review:
0281374
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We derive a posteriori bounds for and its difference quotient , where and are, respectively, the exact and computed solution of a difference approximation to a mildly nonlinear parabolic initial boundary problem, with a known steadystate solution. It is assumed that the computation is over a long interval of time. The estimates are valid for a class of difference approximations, which includes the CrankNicolson method, and are of the same magnitude for both and .
 [1]
Alfred
Carasso, Finitedifference methods and the
eigenvalue problem for nonselfadjoint SturmLiouville operators,
Math. Comp. 23 (1969), 717–729. MR 0258291
(41 #2938), http://dx.doi.org/10.1090/S00255718196902582917
 [2]
Alfred
Carasso and Seymour
V. Parter, An analysis of “boundaryvalue
techniques” for parabolic problems., Math. Comp. 24 (1970), 315–340. MR 0284019
(44 #1249), http://dx.doi.org/10.1090/S00255718197002840199
 [3]
Alfred
Carasso, Longrange numerical solution of mildly nonlinear
parabolic equations., Numer. Math. 16 (1970/1971),
304–321. MR 0286301
(44 #3514)
 [4]
Jim
Douglas Jr., A survey of numerical methods for parabolic
differential equations, Advances in Computers, Vol. 2, Academic
Press, New York, 1961, pp. 1–54. MR 0142211
(25 #5604)
 [5]
Avner
Friedman, Partial differential equations of parabolic type,
PrenticeHall Inc., Englewood Cliffs, N.J., 1964. MR 0181836
(31 #6062)
 [6]
Fritz
John, On integration of parabolic equations by difference methods.
I. Linear and quasilinear equations for the infinite interval, Comm.
Pure Appl. Math. 5 (1952), 155–211. MR 0047885
(13,947b)
 [7]
H. O. Kreiss & O. B. Widlund, Difference Approximations for Initial Value Problems for Partial Differential Equations, Department of Computer Sciences, Report NR 7, Upsala University, 1967.
 [8]
Milton
Lees, Approximate solutions of parabolic equations, J. Soc.
Indust. Appl. Math. 7 (1959), 167–183. MR 0110212
(22 #1092)
 [9]
Robert
D. Richtmyer and K.
W. Morton, Difference methods for initialvalue problems,
Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4,
Interscience Publishers John Wiley & Sons, Inc., New
YorkLondonSydney, 1967. MR 0220455
(36 #3515)
 [10]
O.
B. Widlund, On difference methods for parabolic equations and
alternating direction implicit methods for elliptic equations, IBM J.
Res. Develop. 11 (1967), 239–243. MR 0224312
(36 #7356)
 [1]
 A. Carasso, "Finitedifference methods and the eigenvalue problem for nonselfadjoint SturmLiouville operators," Math. Comp., v. 23, 1969, pp. 717729. MR 0258291 (41:2938)
 [2]
 A. Carasso & S. V. Parter, "An analysis of 'boundaryvalue techniques' for parabolic problems," Math. Comp., v. 24, 1970, pp. 315340. MR 0284019 (44:1249)
 [3]
 A. Carasso, "Long range numerical solution of mildly nonlinear parabolic equations," Numer. Math. (To appear.) MR 0286301 (44:3514)
 [4]
 J. Douglas, Jr., A Survey of Numerical Methods for Parabolic Differential Equations, Advances in Computers, vol. 2, Academic Press, New York, 1961, pp. 154. MR 25 #5604. MR 0142211 (25:5604)
 [5]
 A. Friedman, Partial Differential Equations of Parabolic Type, PrenticeHall, Englewood Cliffs, N. J., 1964. MR 31 #6062. MR 0181836 (31:6062)
 [6]
 F. John, "On integration of parabolic equations by difference methods. I: Linear and quasilinear equations for the infinite interval," Comm. Pure Appl. Math., v. 5, 1952, pp. 155211. MR 13, 947. MR 0047885 (13:947b)
 [7]
 H. O. Kreiss & O. B. Widlund, Difference Approximations for Initial Value Problems for Partial Differential Equations, Department of Computer Sciences, Report NR 7, Upsala University, 1967.
 [8]
 M. Lees, "Approximate solutions of parabolic equations," J. Soc. Indust. Appl. Math., v. 7, 1959, pp. 167183. MR 22 #1092. MR 0110212 (22:1092)
 [9]
 R. D. Richtmyer & K. W. Morton, Difference Methods for InitialValue Problems, 2nd ed., Interscience Tracts in Pure and Appl. Math., no. 4, Interscience, New York, 1967. MR 36 #3515. MR 0220455 (36:3515)
 [10]
 O. B. Widlund, "On difference methods for parabolic equations and alternating direction implicit methods for elliptic equations," IBM J. Res. Develop., v. 11, 1967, pp. 239243. MR 36 #7356. MR 0224312 (36:7356)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65.68
Retrieve articles in all journals
with MSC:
65.68
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197002813740
PII:
S 00255718(1970)02813740
Keywords:
Parabolic equations,
CrankNicolson method,
limiting steady state,
computations over long times
Article copyright:
© Copyright 1970 American Mathematical Society
