Numerical Analysis of a Leapfrog ADI–FDTD Method for Maxwell's Equations in Lossy Media

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Computers and Mathematics with Applications





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Recently, a so-called one-step leapfrog ADI–FDTD method has been developed in engineering community for solving the 3D time-dependent Maxwell’s equations. This method becomes quite popular in simulation wave propagation in graphene-based devices due to its efficiency. We investigate this method from a theoretical point of view by proving the energy conservation property, the unconditional stability of this ADI–FDTD method, and establishing the optimal second-order convergence rate in both time and space on non-uniform cubic grids. Numerical results are presented justifying our analysis.We describe our experience with stenting the patent ductus arteriosusoriginating from the left subclavian artery or innominate artery in patients with a right aortic arch. Patent ductus arteriosus stenting is an alternative to surgical shunt placement in patients with congenital cardiac malformations. Four patients with an elongated patent ductus arteriosus arising from either the left subclavian artery of the innominate artery underwent implantation of multiple stents. The diameter of the stents ranged from 3.5 to 4.5 mm and the length from 8 to 15 mm. Three neonates had tetralogy of Fallot and a right aortic arch, and one 14 month-old patient had discontinuous pulmonary arteries, a right aortic arch, and a ventricular septal defect. In the three neonates with tetralogy of Fallot, we employed an antegrade venous approach. Stent implantation was successful acutely and without complications in all four patients. One patient developed stent restenosisand underwent two additional stent re-dilations 2 months and 5 months after the initial procedure, respectively. The 14 month-old patient underwent planned re-dilation of the stents one month later. All the patients underwent successful complete repair ranging from 149 to 338 days after stent placement. Patent ductus arteriosus stenting in patients with a right aortic arch and a ductus arising from the left subclavian or innominate artery can be performed successfully using multiple stents, particularly in newborns and from a femoral venous approach. Complete repair can be postponed for several months.


Maxwell's equations; Alternating direction implicit method; FDTD method


Applied Mathematics



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