Stable approximations for hyperbolic systems with moving internal boundary conditions
Authors:
M. Goldberg and S. Abarbanel
Journal:
Math. Comp. 28 (1974), 413447
MSC:
Primary 65N10
Corrigendum:
Math. Comp. 29 (1975), 1167.
MathSciNet review:
0381343
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Abstract: The work of Kreiss on the stability theory of difference schemes for the mixed initial boundary value problem for linear hyperbolic systems is extended to deal with the case of the pure initial value problem with an internal boundary. The case of an internal boundary that moves with constant speed c is treated, i.e., . In particular, the stability of "hybrid" schemes is studied by using the LaxWendroff scheme at points that are not on the internal boundary, while using a first order accurate scheme at the internal boundary points. Numerical evidence is given that the results of the linear stability analysis describes the qualitative behavior of such schemes for nonlinear cases, when the internal boundary is a shock.
 [1]
S. Abarbanel & M. Goldberg, "Numerical solution of quasiconservative hyperbolic systemsthe cylindrical shock problem," J. Comput. Phys., v. 10, 1972, pp. 121. MR 0331974 (48:10306)
 [2]
S. Abarbanel & G. Zwas, "An iterative finite difference method for hyperbolic systems," Math. Comp., v. 23, 1969, pp. 549565. MR 40 #1044. MR 0247783 (40:1044)
 [3]
M. Ciment, "Stable matching of difference schemes," SIAM J. Numer. Anal., v. 9, 1972, pp. 695701. MR 0319383 (47:7927)
 [4]
M. Goldberg & S. Abarbanel, A Note on Discontinuities in a Nonlinear Hyperbolic Equation with Piecewise Smooth Data, Dept. of Math. Sciences, Tel Aviv Univ. Report, 1972.
 [5]
M. Goldberg, "A note on the stability of an iterative finitedifference method for hyperbolic systems," Math. Comp., v. 27, 1973, pp. 4144. MR 0341887 (49:6633)
 [6]
B. Gustafsson, H. O. Kreiss & A. Sundström, "Stability theory of difference approximations for mixed initial boundary value problems. II," Math. Comp., v. 26, 1972, pp. 649686. MR 0341888 (49:6634)
 [7]
A. Harten & G. Zwas, "Selfadjusting hybrid schemes for shock computations," J. Comput. Phys., v. 9, 1972, pp. 568583. MR 0309339 (46:8449)
 [8]
H. O. Kreiss, "Difference approximations for the initialboundary value problem for hyperbolic differential equations," Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966), Wiley, New York, 1966, pp. 141166. MR 35 #5156. MR 0214305 (35:5156)
 [9]
H. O. Kreiss, "Stability theory for difference approximations of mixed initial boundary value problems. I," Math. Comp., v. 22, 1968, pp. 703714. MR 39 #2355. MR 0241010 (39:2355)
 [10]
P. D. Lax, "Weak solutions of nonlinear hyperbolic equations and their numerical computations," Comm. Pure Appl. Math., v. 7, 1954, pp. 159193. MR 16, 524. MR 0066040 (16:524g)
 [11]
P. D. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217237. MR 22 #11523. MR 0120774 (22:11523)
 [12]
R. D. Richtmyer & K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed., Interscience Tracts in Pure and Appl. Math., vol. 4, Interscience, New York, 1967. MR 36 #3515. MR 0220455 (36:3515)
 [1]
 S. Abarbanel & M. Goldberg, "Numerical solution of quasiconservative hyperbolic systemsthe cylindrical shock problem," J. Comput. Phys., v. 10, 1972, pp. 121. MR 0331974 (48:10306)
 [2]
 S. Abarbanel & G. Zwas, "An iterative finite difference method for hyperbolic systems," Math. Comp., v. 23, 1969, pp. 549565. MR 40 #1044. MR 0247783 (40:1044)
 [3]
 M. Ciment, "Stable matching of difference schemes," SIAM J. Numer. Anal., v. 9, 1972, pp. 695701. MR 0319383 (47:7927)
 [4]
 M. Goldberg & S. Abarbanel, A Note on Discontinuities in a Nonlinear Hyperbolic Equation with Piecewise Smooth Data, Dept. of Math. Sciences, Tel Aviv Univ. Report, 1972.
 [5]
 M. Goldberg, "A note on the stability of an iterative finitedifference method for hyperbolic systems," Math. Comp., v. 27, 1973, pp. 4144. MR 0341887 (49:6633)
 [6]
 B. Gustafsson, H. O. Kreiss & A. Sundström, "Stability theory of difference approximations for mixed initial boundary value problems. II," Math. Comp., v. 26, 1972, pp. 649686. MR 0341888 (49:6634)
 [7]
 A. Harten & G. Zwas, "Selfadjusting hybrid schemes for shock computations," J. Comput. Phys., v. 9, 1972, pp. 568583. MR 0309339 (46:8449)
 [8]
 H. O. Kreiss, "Difference approximations for the initialboundary value problem for hyperbolic differential equations," Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966), Wiley, New York, 1966, pp. 141166. MR 35 #5156. MR 0214305 (35:5156)
 [9]
 H. O. Kreiss, "Stability theory for difference approximations of mixed initial boundary value problems. I," Math. Comp., v. 22, 1968, pp. 703714. MR 39 #2355. MR 0241010 (39:2355)
 [10]
 P. D. Lax, "Weak solutions of nonlinear hyperbolic equations and their numerical computations," Comm. Pure Appl. Math., v. 7, 1954, pp. 159193. MR 16, 524. MR 0066040 (16:524g)
 [11]
 P. D. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217237. MR 22 #11523. MR 0120774 (22:11523)
 [12]
 R. D. Richtmyer & K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed., Interscience Tracts in Pure and Appl. Math., vol. 4, Interscience, New York, 1967. MR 36 #3515. MR 0220455 (36:3515)
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DOI:
http://dx.doi.org/10.1090/S0025571819740381343X
PII:
S 00255718(1974)0381343X
Article copyright:
© Copyright 1974
American Mathematical Society
