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4 Choosing the spatial discretization scheme Explicit vs. compact implicit schemes for spatial discretization • Explicit schemes are simple and generally easy to implement; • Compact implicit schemes require more computations per time-step; 30 • For explicit methods we need to take a very small time step for stability; • Compact implicit schemes use a small stencil that simplifies the treatment of outer and interior boundaries. Staggered vs. co-located schemes Gottlieb and Yang [22] and Turkel [56] have shown that a staggered scheme is more accurate and efficient than a co-located scheme for the same order of accuracy.

9 Local Regularization We next consider a local regularization. 19) as the connecting function for the local regularization of ε and 1ε . In order to construct a cubic spline we need at least four nodes. Therefore, the minimal δ is equal to 2∆x. 56 On the other hand δ should be less than the half-distance between the interfaces to avoid overdetermination of ε at the intersected nodes. Hence, in our experiments δ is less than 75cm. In the following two tables we present the error v − vδ(h) in the L2 norm on various grids.

6). Since the RHS is linear, this can be trivially solved at each stage of the Runge-Kutta scheme. 7): ∂Px σy 1 + Px = δPx ∂t ε ε ∂Ex σz ∂Px σx + Px = + Ex ∂t ε ∂t ε Define α = [ 14 , 1 1 , , 3 2 1] – the coefficients of the four-stage Runge-Kutta scheme. 4): Pxi+1 = Pxi + αk ∆t δPx − σy Pxi+1 ε 33 and after rearranging we get Pxi+1 = Pxi + αk ∆t δPx ε ∆t 1 + αk ε σy and similarly Exi+1 = Exi + (Pxi+1 − Pxi ) + αk ∆tσx Pxi 1 + αk ∆tσz Inside the physical domain this reduces to the fourth order compact scheme.

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High-order accurate methods for Maxwell equations by Kashdan E.

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