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Convergence of summation-by-parts finite difference methods for the wave equation
In this paper, we consider finite difference approximations of the second
order wave equation. We use finite difference operators satisfying the
summation-by-parts property to discretize the equation in space. Boundary
conditions and grid interface conditions are imposed by the
simultaneous-approximation-term technique. Typically, the truncation error is
larger at the grid points near a boundary or grid interface than that in the
interior. Normal mode analysis can be used to analyze how the large truncation
error affects the convergence rate of the underlying stable numerical scheme.
If the semi-discretized equation satisfies a determinant condition, two orders
are gained from the large truncation error. However, many interesting second
order equations do not satisfy the determinant condition. We then carefully
analyze the solution of the boundary system to derive a sharp estimate for the
error in the solution and acquire the gain in convergence rate. The result
shows that stability does not automatically yield a gain of two orders in
convergence rate. The accuracy analysis is verified by numerical experiments.Comment: In version 2, we have added a new section on the convergence analysis
of the Neumann problem, and have improved formulations in many place
High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension
The nonlinear Helmholtz equation (NLH) models the propagation of
electromagnetic waves in Kerr media, and describes a range of important
phenomena in nonlinear optics and in other areas. In our previous work, we
developed a fourth order method for its numerical solution that involved an
iterative solver based on freezing the nonlinearity. The method enabled a
direct simulation of nonlinear self-focusing in the nonparaxial regime, and a
quantitative prediction of backscattering. However, our simulations showed that
there is a threshold value for the magnitude of the nonlinearity, above which
the iterations diverge. In this study, we numerically solve the one-dimensional
NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity
contains absolute values of the field, the NLH has to be recast as a system of
two real equations in order to apply Newton's method. Our numerical simulations
show that Newton's method converges rapidly and, in contradistinction with the
iterations based on freezing the nonlinearity, enables computations for very
high levels of nonlinearity. In addition, we introduce a novel compact
finite-volume fourth order discretization for the NLH with material
discontinuities.The one-dimensional results of the current paper create a
foundation for the analysis of multi-dimensional problems in the future.Comment: 47 pages, 8 figure
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