Abstract
Numerical approximation of wave propagation can be done very efficiently on uniform grids. The Yee scheme is a good example. A serious problem with uniform grids is the approximation of boundary conditions at a boundary not aligned with the grid. In this paper, boundary conditions are introduced by modifying appropriate material coefficients at a few grid points close to the embedded boundary. This procedure is applied to the Yee scheme and the resulting method is proven to be \(L^2\)-stable in one space dimension. Depending on the boundary approximation technique it is of first or second order accuracy even if the boundary is located at an arbitrary point relative to the grid. This boundary treatment is applied also to a higher order discretization resulting in a third order accurate method. All algorithms have the same staggered grid structure in the interior as well as across the boundaries for efficiency. A numerical example with the extension to two space dimensions is included.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Abenius, U. Andersson, F. Edelvik, L. Eriksson, and G. Ledfelt, Hybrid time domain solvers for the Maxwell equations in 2D, Int. J. Numer. Methods Eng., 53 (2002), pp. 2185–2199.
G. Chesshire and W. Henshaw, Composite overlapping meshes for the solution of partial differential equations, J. Comput. Phys., 90 (1990), pp. 1–64.
G. Cohen and P. Joly, Construction analysis of fourth-order difference schemes for the acoustic wave equation in nonhomogeneous media, SIAM J. Numer. Anal., 33 (1996), pp. 1266–1302.
B. Fornberg, High-order finite differences and the pseudospectral method on staggered grids, SIAM J. Numer. Anal., 27 (1990), pp. 904–918.
B. Fornberg, M. Ghrist, and T. Driscoll, Staggered time integrators for wave equations, SIAM J. Numer. Anal., 38 (2000), pp. 718–741.
B. Gustafsson, The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comput., 29 (1975), pp. 396–406.
B. Gustafsson, The convergence rate for difference approximations to general mixed initial boundary value problems, SIAM J. Numer. Anal., 18 (1981), pp. 179–190.
B. Gustafsson and E. Mossberg, Time compact high order difference methods for wave propagation, SIAM J. Sci. Comput., 26 (2004), pp. 259–271.
B. Gustafsson and P. Wahlund, Time compact difference methods for wave propagation in discontinuous media, SIAM J. Sci. Comput., 26 (2004), pp. 272–293.
B. Gustafsson and P. Wahlund, Time compact high order difference methods for wave propagation, 2-d, J. Sci. Comput., 25 (2005), pp. 195–211.
C. Helzel, M. Berger, and R. LeVeque, A high-resolution rotated grid method for conservation laws with embedded geometries, SIAM J. Sci. Comput., 26 (2005), pp. 785–809.
H.-O. Kreiss, N. Petersson, and J. Yström, Difference approximations of the Neumann problem for the second order wave equation, SIAM J. Numer. Anal., 42 (2004), pp. 1292–1323.
P. Lax and B. Wendroff, Difference schemes for hyperbolic equations with high order accuracy, Comm. Pure Appl. Math., 17 (1964), pp. 381–398.
G. Shubin and J. Bell, A modified equation approach to constructing fourth order methods for acoustic wave propagation, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 135–151.
A. Tornberg and B. Engquist, Regularization techniques for numerical approximation of PDEs with singularities, J. Sci. Comput., 19 (2003), pp. 527–552.
A. Tornberg and B. Engquist, Numerical approximations of singular source terms in differential equations, J. Comput. Phys., 200 (2004), pp. 462–488.
A. Tornberg and B. Engquist, High order difference methods for wave propagation in discontinuous media, to appear (2006).
J. Tuomela, On the construction of arbitrary high order schemes for the many-dimensional wave equation, BIT, 36 (1996), pp. 158–165.
E. Turkel and A. Yefet, On the construction of a high order difference scheme for complex domains in a cartesian grid, Appl. Numer. Anal., 33 (2000), pp. 113–124.
K. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag., 14 (1966), pp. 302–307.
Author information
Authors and Affiliations
Corresponding author
Additional information
AMS subject classification (2000)
65M06, 65M12
Rights and permissions
About this article
Cite this article
Tornberg, AK., Engquist, B., Gustafsson, B. et al. A new type of boundary treatment for wave propagation . Bit Numer Math 46 (Suppl 1), 145–170 (2006). https://doi.org/10.1007/s10543-006-0088-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-006-0088-6