Stable difference schemes with uneven mesh spacings
Abstract: We consider a finite-difference approximation to the Cauchy problem for a firstorder hyperbolic partial differential equation using different mesh spacings in different portions of the domain. By reformulating our problem as a difference approximation to an initial-boundary value problem, we are able to use the theory of H. O. Kreiss and S. Osher to analyze the stability of our scheme.
-  M. Ciment, Stable Difference Schemes With Uneven Mesh Spacings, A.E.C. Research and Development Report #NYO-1480-100, Ph.D. Thesis, New York University, New York, 1968.
-  Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
-  Heinz-Otto Kreiss, Difference approximations for the initial-boundary value problem for hyperbolic differential equations, Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966) John Wiley & Sons, Inc., New York, 1966, pp. 141–166. MR 0214305
-  Heinz-Otto Kreiss, Stability theory for difference approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714. MR 0241010, https://doi.org/10.1090/S0025-5718-1968-0241010-7
-  Peter D. Lax and Burton Wendroff, Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math. 17 (1964), 381–398. MR 0170484, https://doi.org/10.1002/cpa.3160170311
-  Stanley Osher, Systems of difference equations with general homogeneous boundary conditions, Trans. Amer. Math. Soc. 137 (1969), 177–201. MR 0237982, https://doi.org/10.1090/S0002-9947-1969-0237982-4
-  Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
- M. Ciment, Stable Difference Schemes With Uneven Mesh Spacings, A.E.C. Research and Development Report #NYO-1480-100, Ph.D. Thesis, New York University, New York, 1968.
- E. Isaacson & H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. MR 34 # 924. MR 0201039 (34:924)
- H. O. Kreiss, Difference Approximations for Initial-Boundary Value Problems for Hyperbolic Differential Equations, Proc. Adv. Sympos. Numerical Solutions of Partial Differential Equations (Madison, Wis., 1965), Wiley, New York, 1966, pp. 141-166. MR 35 #5156. MR 0214305 (35:5156)
- H. O. Kreiss, "Stability theory for difference approximations of mixed initial boundary value problems. I," Math. Comp., v. 22, 1968, pp. 703-714. MR 39 #2355. MR 0241010 (39:2355)
- P. D. Lax & B. Wendroff, "Difference schemes for hyperbolic equations with higher order accuracy," Comm. Pure Appl. Math., v. 17, 1964, 381-398. MR 30 #722. MR 0170484 (30:722)
- S. Osher, "Systems of difference equations with general homogeneous boundary conditions," Trans. Amer. Math. Soc., v. 137, 1969, pp. 177-201. MR 38 #6259. MR 0237982 (38:6259)
- R. D. Richtmyer & K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed., Interscience Tracts in Pure and Appl. Math., no. 4, Interscience, New York, 1967. MR 36 #3515. MR 0220455 (36:3515)
Retrieve articles in Mathematics of Computation with MSC: 65N10
Retrieve articles in all journals with MSC: 65N10
Keywords: Difference methods, stability, mixed initial-boundary value problems, mesh refinement
Article copyright: © Copyright 1971 American Mathematical Society