Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Stability of interfaces with mesh refinement

Author: Marsha J. Berger
Journal: Math. Comp. 45 (1985), 301-318
MSC: Primary 65M10; Secondary 65M50
MathSciNet review: 804925
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the stability of mesh refinement in space and time for several different interface equations and finite-difference approximations. First, we derive a root condition which implies stability for the initial-boundary 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 Lax-Wendroff with all the interface conditions considered, and for Leapfrog with interpolation interface conditions when the fine and coarse grids overlap.

References [Enhancements On Off] (What's this?)

  • [1] L. Ahlfors, Complex Analysis, McGraw-Hill, 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. 84-43, September, 1984.
  • [4] M. Berger & J. Oliger, "Adaptive mesh refinement for hyperbolic partial differential equations," J. Comput. Phys., v. 53, 1984, pp. 484-512. 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. 80-0130, January 1980.
  • [7] G. Browning, H.-O. Kreiss & J. Oliger, "Mesh refinement," Math. Comp., v. 27, 1973, pp. 29-39. MR 0334542 (48:12861)
  • [8] M. Ciment, "Stable difference schemes with uneven mesh spacings," Math. Comp., v. 25, 1971, pp. 219-226. MR 0300470 (45:9516)
  • [9] M. Ciment, "Stable matching of difference schemes," SIAM J. Numer. Anal., v. 9, 1972, pp. 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] M. Goldberg & E. Tadmor, "Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II," Math. Comp., v. 36, 1981, pp. 603-626. MR 606519 (83f:65142)
  • [12] W. Gropp, "A test of moving mesh refinement for 2-D scalar hyperbolic problems," SIAM J. Sci. Statist. Comput., v. 1, 1980, pp. 191-197. MR 594754 (82j:65088)
  • [13] B. Gustafsson, Proc. 1983 AMS-SIAM 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. 649-686. MR 0341888 (49:6634)
  • [15] B. Kreiss, "Construction of curvilinear grids," SIAM J. Sci. Statist. Comput., v. 4, 1983, pp. 270-279. 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. 703-714. 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. 177-201. 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. 1-16. 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. 241-255. 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. 199-217. 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

Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society