We consider the full set of electromagnetic fields with azimuthal symmetry. Many of the ideas for the electromagnetic field representation and solution can trace roots to Langdon  and Eastwood . Maxwell's equations in integral form are written:
We define a new set of field variables which encapsulate the mesh metrics:
In Eqs. (1.2), the line integrals are along cell sides, and the surface integrals are over surfaces formed by cell faces, where fields are as defined on the Yee mesh . The constituency equations for the integral-form variables become
where and are coupling matrices with the dimensionality of capacitance and inductance, respectively. On a general mesh, the couplings can involve fields from one or more neighboring cells; the general mesh is beyond the scope of the present paper. On a non-uniform orthogonal Yee mesh in cylindrical z-r coordinates, the capacitances in OOPIC are written:
Similarly, the inductance elements for a non-uniform orthogonal Yee mesh in cylindrical z-r coordinates in OOPIC are:
The Maxwell curl equations can be written in terms of these variables and discretized. The transverse magnetic (TM) set becomes:
where over the cell volume. The indices refer to locations on the Yee mesh, and are ordered in z, r for a right-handed cylindrical coordinate system. The corresponding transverse electric (TE) set can be written:
The TM and TE field equations are advanced in time using a leap frog advance . The source terms in Eqs. (6) and (7), I, are the currents resulting from charged particle motion. The field algorithms are completed by initial conditions and boundary conditions.