Higher Order FDTD Schemes in Curvilinear Coordinates

6 HIGHER ORDER FDTD SCHEMES IN CURVILINEAR COORDINATES 

If the above transformation is also orthogonal, namely = 0 and F 0, then (2.131) becomes 

In this manner, the algorithm exploits the properties of lattice orthogonality, such as the zero off-diagonal elements in the metric tensors, which lessen the overhead of the computations. 

To construct the fourth-order FDTD scheme, let us apply the Taylor expansion series to Hu (similarly for the other components) as in (2.14), and obtain 

The next concern lies on the computation of the term d3 /dt3 in (2.134). Taking the third-order temporal derivative of (2.129), one gets 

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