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The Method of Lines (MoL) in combination with impedance/admittance and field transformation (IAFT) is used to analyze electromagnetic waves. The used cases are waveguiding structures in microwave technology and optics. The core of the theory is the solution of generalized transmission line equations (GTL). In the case of complex structures, a combination with finite differences (FD) can be used. The quality of this solution essentially depends on the effectiveness of the used interpolation of the differences. The individual steps of the FD are permanently linked to the steps of the fully vectorial impedance/admittance and field transformation, so standard libraries cannot be used. Two approaches based on the linear and quadratic interpolation were built into the impedance/admittance and field transformation in the past. However, the degree of improvement due to one or another kind of interpolation depends on the concrete behavior of the solution sought. In the case of complex structures, choosing the appropriate type of interpolation should be an effective aid. In this paper, an extension of the family of built-in methods is proposed - with the possibility of being able to build any known numerical method from the class of one-step or multi-step methods into the GTL solution. These can be higher-order methods, including fast explicit methods, or particularly stable implicit methods. The transmission matrices for the impedance/admittance and field transformation serve as the building site. To illustrate the procedure, some different methods are integrated into the GTL solution. The efficiency of the solutions is tested on some test structures and compared with each other and with existing solutions. The relevant waveguide specifics were discussed. Initial systematics and recommendations for users were derived.

The Method of Lines is a semi-analytical versatile tool for the solution of partial differential equations. For the analysis of spatial complex linear waveguide structures, this method is combined with impedance/admittance and field transformation, as well as with finite differences. This paper extends this approach to the treatment of structures with non-linear dielectric materials. The non-linear generalized transmission line equations are derived. An iterative algorithm based on the impedance/admittance transformation with the field transformation obtains efficient and self-consistent solutions. For demonstration, a non-linear stripe waveguide is considered. The Kerr non-linearity was investigated, though the general case is treatable. As a criterion for the correctness of the algorithm, a second harmonic generation as well as a bidirectional, spatially, and temporally energy exchange between the harmonics was examined. The specific limiting factors for the algorithm were explored. The approach can be used for any spatial structure, including, for instance, photonic crystal waveguides and metamaterials.

The Method of Lines in combination with impedance/admittance and field transformation is used to analyze electromagnetic waves. The used cases are waveguiding structures in microwave technology and optics. The core of the theory is the solution of generalized transmission line equations (GTL). In the case of complex structures, a combination with finite differences (FD) can be used. The quality of this solution essentially depends on the effectiveness of the used interpolation of the differences. The individual steps of the FD are permanently linked to the steps of the fully vectorial impedance/admittance and field transformation, so standard libraries cannot be used. Two approaches based on the linear and quadratic interpolation were built into the impedance/admittance and field transformation in the past. However, the degree of improvement due to one or another kind of interpolation depends on the concrete behavior of the solution sought. In the case of complex structures, choosing the appropriate type of interpolation should be an effective aid. In this paper, an extension of the family of built-in methods is proposed—with the possibility of being able to build any known numerical method from the class of one-step or multi-step methods into the GTL solution. These can be higher-order methods, including fast explicit methods, or particularly stable implicit methods. The transmission matrices for the impedance/admittance and field transformation serve as the building site. To illustrate the procedure, some different methods are integrated into the GTL solution. The efficiency of the solutions is tested on some test structures and compared with each other and with existing solutions. The relevant waveguide specifics were discussed. Initial systematics and recommendations for users were derived.

The Method of Lines (MoL) in combination with impedance-/admittance and field transformation (IAFT) is used to analyze electromagnetic waves. The used cases are wave-guiding structures in microwave technology and optics. The core of the theory is the solution of generalized transmission line equations (GTL). In the case of complex structures, a combination with finite differences (FD) can be used. The quality of this solution essentially depends on the effectiveness of the used interpolation of the differences. The individual steps of the FD are permanently linked to the steps of the fully vectorial impedance-/admittance and/or field transformation, so standard libraries cannot be used. Two approaches based on the linear and quadratic interpolation were built into the impedance-/admittance and field transformation in the past. However, the degree of improvement due to one or another kind of interpolation depends on the concrete behavior of the solution sought. In the case of complex structures, choosing the appropriate type of interpolation should be an effective aid. In this paper, an extension of the family of built-in methods is proposed - with the possibility of being able to build any known numerical method from the class of one-step or multi-step methods into the GTL solution. These can be higher-order methods, including fast explicit methods, or particularly stable implicit methods. The transmission matrices for the impedance-/admittance and field transformation serve as the building site. To illustrate the procedure, some different methods are integrated into the GTL solution. The accuracy of the solutions is tested on selected complex structures and compared with each other and with existing solutions. It is shown that the optimal choice of method and the quality of the solution can depend on concrete structures. Keywords: Method of lines, generalized transmission line equations, impedance/admittance transformation, waveguide structures, finite differences, finite differences with second order accuracy.