By Marnaghan F. D., Wintner A.

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**Example text**

Never the less, based on [47], we expect the global error to be higher than second order. 6 Discrete regularization Model problem 1 is mainly of mathematical importance. Hence, we introduce model problem 2, which can be considered as a model for the electromagnetic wave scattering by a physical body. 17). A natural way to approximate the piecewise continuous function ε(x) is by using a high order polynomial interpolation. We do not require higher than second order of differentiability for the regularized function, because the analytic solution has only one continuous derivative at the interface.

1) by t, differentiation of the second equation by x and its substitution into the first equation. For both model problems the solution has the form E(x, t) = u(x)eiωt , where u(x) satisfies the Helmholtz equation: uxx + [ω 2 Q(x)]u = 0 This is a second order linear elliptic equation with variable coefficients, where Q(x) = ε(x)µ = 1 . 3) For both model problems the solution has the form H(x, t) = uˆ(x)eiωt , where uˆ satisfies the second order ODE: 1 ∂ 1 ∂ uˆ µ ∂x ε(x) ∂x + ω 2 uˆ = 0 For piecewise-constant coefficients we can find an explicit solution of the Helmholtz equation for both model problems.

These schemes are divided into two classes: explicit schemes and compact implicit schemes. Each class has its own subdivision into the staggered and co-located schemes. In order to establish the notation, we write the second-order accurate spatial derivative operator as (Du)i = (Du)i ≈ ∂u(xi ) ∂x ui+ 1 − ui− 1 2 2 ∆x with a local truncation error of order O(∆x2 ). 3) The fourth-order explicit scheme for the spatial discretization of Maxwell equations was discussed by Taflove [50]. 3) for numerical solution of the Maxwell equations on unbounded domains.

### A Canonical Form for Real Matrices under Orthogonal Transformations by Marnaghan F. D., Wintner A.

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