Finite difference discretizations of some initial and boundary value problems with interface

Akrivis, Georgios D.; Dougalis, Vassilios A.

doi:10.1090/S0025-5718-1991-1066829-7

Finite difference discretizations of some initial and boundary value problems with interface
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by Georgios D. Akrivis and Vassilios A. Dougalis PDF

Math. Comp. 56 (1991), 505-522 Request permission

Abstract:

We analyze the discretization of initial and boundary value problems with a stationary interface in one space dimension for the heat equation, the Schrödinger equation, and the wave equation by finite difference methods. Extending the concept of the elliptic projection, well known from the analysis of Galerkin finite element methods, to our finite difference case, we prove second-order error estimates in space and time in the ${l^2}$ norm.

References

G. D. Akrivis and V. A. Dougalis, On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 6, 643–670 (English, with French summary). MR 1135988, DOI 10.1051/m2an/1991250606431
Laurence A. Bales, Vassilios A. Dougalis, and Steven M. Serbin, Cosine methods for second-order hyperbolic equations with time-dependent coefficients, Math. Comp. 45 (1985), no. 171, 65–89. MR 790645, DOI 10.1090/S0025-5718-1985-0790645-1
B. M. Budak, On homogeneous differential-difference schemes of second-order accuracy for parabolic and hyperbolic equations with discontinuous coefficients, Vestnik Moskov. Univ. Ser. I Mat. Meh. 1962 (1962), no. 2, 7–13 (Russian, with English summary). MR 0144483
Eugene C. Gartland Jr., Compact high-order finite differences for interface problems in one dimension, IMA J. Numer. Anal. 9 (1989), no. 2, 243–260. MR 1000460, DOI 10.1093/imanum/9.2.243
Eugene Isaacson, Error estimates for parabolic equations, Comm. Pure Appl. Math. 14 (1961), 381–389. MR 137311, DOI 10.1002/cpa.3160140315
Suzanne T. McDaniel, Applications of energy methods to finite-difference solutions of the parabolic wave equation, Comput. Math. Appl. 11 (1985), no. 7-8, 823–829. Computational ocean acoustics (New Haven, Conn., 1984). MR 809614, DOI 10.1016/0898-1221(85)90177-4
Suzanne T. McDaniel and Ding Lee, A finite-difference treatment of interface conditions for the parabolic wave equation: the horizontal interface, J. Acoust. Soc. Amer. 71 (1982), no. 4, 855–858. MR 651838, DOI 10.1121/1.387611

A priori estimates for difference equations

1

Homogeneous difference schemes on non-uniform nets for equations of parabolic type

3

A. A. Samarskii and I. V. Frjazinov, On the convergence of difference schemes for a heat-conduction equation with discontinuous coefficients, U.S.S.R. Comput. Math. and Math. Phys. 4 (1962), 962–982. MR 0205493
A. A. Samarskij, Theorie der Differenzenverfahren, Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], vol. 40, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1984 (German). Translated from the Russian by Gisbert Stoyan. MR 783639
Vidar Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR 744045
Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
Mary Fanett Wheeler, A priori $L_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723–759. MR 351124, DOI 10.1137/0710062
Mary Fanett Wheeler, $L_{\infty }$ estimates of optimal orders for Galerkin methods for one-dimensional second order parabolic and hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 908–913. MR 343658, DOI 10.1137/0710076
Mary Fanett Wheeler, An optimal $L_{\infty }$ error estimate for Galerkin approximations to solutions of two-point boundary value problems, SIAM J. Numer. Anal. 10 (1973), 914–917. MR 343659, DOI 10.1137/0710077