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The upwind leapfrog or Linear Bicharacteristic Scheme (LBS) has previously been extended to treat lossy dielectric and lossy magnetic materials. This report extends the Linear Bicharacteristic Scheme for computational electromagnetics to the two-dimensional case, which includes treatment of lossy dielectric and magnetic materials and perfect electrical conductors. This is accomplished by implementing the LBS for homogeneous lossy dielectric and magnetic media and for perfect electrical conductors. Heterogeneous media are modeled by applying surface boundary conditions, and no special extrapolations or interpolations at dielectric material boundaries are required. The Perfectly Matched Layer (PML) outer boundary concept is also developed for this scheme. Results are presented for two-dimensional model problems on uniform grids, and the FDTD algorithm is chosen as a convenient reference algorithm for comparison. The results demonstrate that the explicit LBS is a dissipation-free, second-order accurate algorithm which uses an upwind computational stencil rather than a central difference stencil, and yet it has approximately one-third the phase velocity error. Computational requirements are also discussed.