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A new type of boundary treatment for wave propagation. (English) Zbl 1106.65078
Summary: Numerical approximation of wave propagation can be done very efficiently on uniform grids. The Yee scheme is a good example [cf. K. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag. 14, 302–307 (1966)]. 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.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Software:
CMPGRD
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