The Topologically Consistent Higher Order Nonstandard Framework

2 The Topologically Consistent Higher Order Nonstandard Framework 

Comprehending the role of reflection errors in nonorthogonal domains and the defects of the conventional higher order forms, a class of higher order nonstandard curvilinear schemes for the approximation of space and time derivatives in Maxwell s equations has been introduced. The principal attribute focuses on the representation of fields by a family of robust models that involve a significantly increased amount of lattice points in the vicinity of the central node [33, 36] as opposed to existing techniques. 

Consider a general coordinate system (u, v, w) defined by its g(u,v, zu) metrics which typify all implementation issues and the appropriate grid selection. Assuming that all derivatives throughout the formulation will be evaluated with an Mth-order accuracy, the parametric expressions for the spatial and temporal approximants, respectively, are given by 

Proceeding to (3.70), special attention must be drawn to the multidirectional operator Qfz, [ ] whose extra degree of freedom L denotes the suitable stencil size, l At, along each lattice direction with a regular value of Z, = 3. Its generalized definition in terms of g(u, v, w) metrics is 

Parameters Ym and improve total robustness in the manipulation of the fine structural details, encountered at a waveguide or an antenna, by the fulfillment of the subsequent Mth-order gauges 

---