Stability of interfaces with mesh refinement
Author:
Marsha J. Berger
Journal:
Math. Comp. 45 (1985), 301318
MSC:
Primary 65M10; Secondary 65M50
MathSciNet review:
804925
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We study the stability of mesh refinement in space and time for several different interface equations and finitedifference approximations. First, we derive a root condition which implies stability for the initialboundary value problem for this type of interface. From the root condition, we prove the stability of several interface equations using the maximum principle. In some cases, the final verification steps can be done analytically; in other cases, a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle. Using this method, we prove stability for LaxWendroff with all the interface conditions considered, and for Leapfrog with interpolation interface conditions when the fine and coarse grids overlap.
 [1]
Lars
V. Ahlfors, Complex analysis, 3rd ed., McGrawHill Book Co.,
New York, 1978. An introduction to the theory of analytic functions of one
complex variable; International Series in Pure and Applied Mathematics. MR 510197
(80c:30001)
 [2]
M. Berger, Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations, Ph. D. dissertation, Computer Science Dept., Stanford University, 1982.
 [3]
M. Berger, On Conservation at Grid Interfaces, Icase Report No. 8443, September, 1984.
 [4]
Marsha
J. Berger and Joseph
Oliger, Adaptive mesh refinement for hyperbolic partial
differential equations, J. Comput. Phys. 53 (1984),
no. 3, 484–512. MR 739112
(85h:65211), http://dx.doi.org/10.1016/00219991(84)900731
 [5]
J. Bolstad, An Adaptive Finite Difference Method for Hyperbolic Systems in One Space Dimension, Ph. D. dissertation, Computer Science Dept., Stanford University, 1982.
 [6]
C. W. Boppe & M. A. Stern, Simulated Transonic Flows for Aircraft with Nacelles, Pylons, and Winglets, AIAA Paper No. 800130, January 1980.
 [7]
Gerald
Browning, HeinzOtto
Kreiss, and Joseph
Oliger, Mesh refinement, Math. Comp. 27 (1973), 29–39. MR 0334542
(48 #12861), http://dx.doi.org/10.1090/S00255718197303345426
 [8]
Melvyn
Ciment, Stable difference schemes with uneven
mesh spacings, Math. Comp. 25 (1971), 219–227. MR 0300470
(45 #9516), http://dx.doi.org/10.1090/S00255718197103004703
 [9]
Melvyn
Ciment, Stable matching of difference schemes, SIAM J. Numer.
Anal. 9 (1972), 695–701. MR 0319383
(47 #7927)
 [10]
W. Coughran, On the Approximate Solution of Hyperbolic Initial Boundary Value Problems, Ph. D. dissertation, Computer Science Dept., Stanford University, 1980.
 [11]
Moshe
Goldberg and Eitan
Tadmor, Schemeindependent stability criteria
for difference approximations of hyperbolic initialboundary value
problems. II, Math. Comp.
36 (1981), no. 154, 603–626. MR 606519
(83f:65142), http://dx.doi.org/10.1090/S00255718198106065199
 [12]
William
D. Gropp, A test of moving mesh refinement for 2D scalar
hyperbolic problems, SIAM J. Sci. Statist. Comput. 1
(1980), no. 2, 191–197. MR 594754
(82j:65088), http://dx.doi.org/10.1137/0901012
 [13]
Bertil
Gustafsson, Numerical boundary conditions, Largescale
computations in fluid mechanics, Part 1 (La Jolla, Calif., 1983), Lectures
in Appl. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1985,
pp. 279–308. MR 818773
(87b:65145)
 [14]
Bertil
Gustafsson, HeinzOtto
Kreiss, and Arne
Sundström, Stability theory of difference
approximations for mixed initial boundary value problems. II, Math. Comp. 26 (1972), 649–686. MR 0341888
(49 #6634), http://dx.doi.org/10.1090/S00255718197203418883
 [15]
Barbro
Kreiss, Construction of a curvilinear grid, SIAM J. Sci.
Statist. Comput. 4 (1983), no. 2, 270–279. MR 697180
(84e:65090), http://dx.doi.org/10.1137/0904021
 [16]
HeinzOtto
Kreiss, Stability theory for difference
approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714. MR 0241010
(39 #2355), http://dx.doi.org/10.1090/S00255718196802410107
 [17]
J. Oliger, personal communication, 1979.
 [18]
Stanley
Osher, Systems of difference equations with
general homogeneous boundary conditions, Trans.
Amer. Math. Soc. 137 (1969), 177–201. MR 0237982
(38 #6259), http://dx.doi.org/10.1090/S00029947196902379824
 [19]
Victor
Pereyra, Wlodzimierz
Proskurowski, and Olof
Widlund, High order fast Laplace solvers for
the Dirichlet problem on general regions, Math.
Comp. 31 (1977), no. 137, 1–16. MR 0431736
(55 #4731), http://dx.doi.org/10.1090/S0025571819770431736X
 [20]
L. Reyna, On Composite Meshes, Ph. D. dissertation, Applied Math Dept., California Institute of Technology, 1983.
 [21]
Göran
Starius, On composite mesh difference methods for hyperbolic
differential equations, Numer. Math. 35 (1980),
no. 3, 241–255. MR 592156
(82b:65089), http://dx.doi.org/10.1007/BF01396411
 [22]
L. N. Trefethen, Wave Propagation and Stability for Finite Difference Schemes, Ph. D. dissertation, Computer Science Dept., Stanford University, 1982.
 [23]
Lloyd
N. Trefethen, Group velocity interpretation of the stability theory
of Gustafsson, Kreiss, and Sundström, J. Comput. Phys.
49 (1983), no. 2, 199–217. MR 699214
(84e:65096), http://dx.doi.org/10.1016/00219991(83)901237
 [24]
Robert
Vichnevetsky and John
B. Bowles, Fourier analysis of numerical approximations of
hyperbolic equations, SIAM Studies in Applied Mathematics,
vol. 5, Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, Pa., 1982. With a foreword by Garrett Birkhoff. MR 675265
(84f:65002)
 [1]
 L. Ahlfors, Complex Analysis, McGrawHill, New York, 1979. MR 510197 (80c:30001)
 [2]
 M. Berger, Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations, Ph. D. dissertation, Computer Science Dept., Stanford University, 1982.
 [3]
 M. Berger, On Conservation at Grid Interfaces, Icase Report No. 8443, September, 1984.
 [4]
 M. Berger & J. Oliger, "Adaptive mesh refinement for hyperbolic partial differential equations," J. Comput. Phys., v. 53, 1984, pp. 484512. MR 739112 (85h:65211)
 [5]
 J. Bolstad, An Adaptive Finite Difference Method for Hyperbolic Systems in One Space Dimension, Ph. D. dissertation, Computer Science Dept., Stanford University, 1982.
 [6]
 C. W. Boppe & M. A. Stern, Simulated Transonic Flows for Aircraft with Nacelles, Pylons, and Winglets, AIAA Paper No. 800130, January 1980.
 [7]
 G. Browning, H.O. Kreiss & J. Oliger, "Mesh refinement," Math. Comp., v. 27, 1973, pp. 2939. MR 0334542 (48:12861)
 [8]
 M. Ciment, "Stable difference schemes with uneven mesh spacings," Math. Comp., v. 25, 1971, pp. 219226. MR 0300470 (45:9516)
 [9]
 M. Ciment, "Stable matching of difference schemes," SIAM J. Numer. Anal., v. 9, 1972, pp. 695701. MR 0319383 (47:7927)
 [10]
 W. Coughran, On the Approximate Solution of Hyperbolic Initial Boundary Value Problems, Ph. D. dissertation, Computer Science Dept., Stanford University, 1980.
 [11]
 M. Goldberg & E. Tadmor, "Schemeindependent stability criteria for difference approximations of hyperbolic initialboundary value problems. II," Math. Comp., v. 36, 1981, pp. 603626. MR 606519 (83f:65142)
 [12]
 W. Gropp, "A test of moving mesh refinement for 2D scalar hyperbolic problems," SIAM J. Sci. Statist. Comput., v. 1, 1980, pp. 191197. MR 594754 (82j:65088)
 [13]
 B. Gustafsson, Proc. 1983 AMSSIAM Summer Seminar on Large Scale Computations in Fluid Mechanics (La Jolla California), Lectures in Applied Math., vol. 22, Amer. Math. Soc., Providence, R. I., 1985. MR 818773 (87b:65145)
 [14]
 B. Gustafsson, H.O. Kreiss & A. Sundström, "Stability theory of difference approximations for initial boundary value problems. II," Math. Comp., v. 26, 1972, pp. 649686. MR 0341888 (49:6634)
 [15]
 B. Kreiss, "Construction of curvilinear grids," SIAM J. Sci. Statist. Comput., v. 4, 1983, pp. 270279. MR 697180 (84e:65090)
 [16]
 H.O. Kreiss, "Stability theory for difference approximations of mixed initial boundary value problems. I," Math. Comp., v. 22, 1968, pp. 703714. MR 0241010 (39:2355)
 [17]
 J. Oliger, personal communication, 1979.
 [18]
 S. Osher, "Systems of difference equations with general homogeneous boundary conditions," Trans. Amer. Math. Soc., v. 137, 1969, pp. 177201. MR 0237982 (38:6259)
 [19]
 V. Pereyra, W. Proskurowski & O. Widlund, "High order fast Laplace solvers for the Dirichlet problem on general regions," Math. Comp., v. 31, 1977, pp. 116. MR 0431736 (55:4731)
 [20]
 L. Reyna, On Composite Meshes, Ph. D. dissertation, Applied Math Dept., California Institute of Technology, 1983.
 [21]
 G. Starius, "On composite mesh difference methods for hyperbolic differential equations," Numer. Math., v. 35, 1980, pp. 241255. MR 592156 (82b:65089)
 [22]
 L. N. Trefethen, Wave Propagation and Stability for Finite Difference Schemes, Ph. D. dissertation, Computer Science Dept., Stanford University, 1982.
 [23]
 L. N. Trefethen, "Group Velocity Interpretation of the Stability Theory of Gustafsson, Kreiss and Sundström." J. Comput. Phys., v. 49, 1983, pp. 199217. MR 699214 (84e:65096)
 [24]
 R. Vichnevetsky & J. Bowles, Fourier Analysis of Numerical Approximations of Hyperbolic Equations, SIAM, Philadelphia, Pa., 1982. MR 675265 (84f:65002)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65M10,
65M50
Retrieve articles in all journals
with MSC:
65M10,
65M50
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198508049254
PII:
S 00255718(1985)08049254
Article copyright:
© Copyright 1985 American Mathematical Society
