Finite difference techniques are used to solve a variety of differential equations. For the neutron diffusion equation, the typical local truncation error for standard finite difference approximation is on the order of the mesh spacing squared. To improve the accuracy of the finite difference approximation of the diffusion equation, the invariance properties of the original differential equation have been incorporated into the finite difference equations. Using the concept of an invariant difference operator, the invariant difference approximations of the multi-group neutron diffusion equation were determined in one-dimensional slab and two-dimensional Cartesian coordinates, for multiple region problems. These invariant difference equations were defined to lie upon a cell edged mesh as opposed to the standard difference equations, which lie upon a cell centered mesh. Results for a variety of source approximations showed that the invariant difference equations were able to determine the eigenvalue with greater accuracy, for a given mesh spacing, than the standard difference approximation. The local truncation errors for these invariant difference schemes were found to be highly dependent upon the source approximation used, and the type of source distribution played a greater role in determining the accuracy of the invariant difference scheme than the local truncation error.

Experiments were performed to characterize the creation and evolution of high-temperature (T{sub e}{approximately}100eV), high-density (n{sub e}>10{sup 22}cm{sup {minus}3}) plasmas created with intense ({approximately}10{sup 12}-10{sup 16}W/cm{sup 2}), ultra-short (130fs) laser pulses. The principle diagnostic was plasma reflectivity at optical wavelengths (614nm). An array of target materials (Al, Au, Si, SiO{sub 2}) with widely differing electronic properties tested plasma behavior over a large set of initial states. Time-integrated plasma reflectivity was measured as a function of laser intensity. Space- and time-resolved reflectivity, transmission and scatter were measured with a spatial resolution of {approximately}3{mu}m and a temporal resolution of 130fs. An amplified, mode-locked dye laser system was designed to produce {approximately}3.5mJ, {approximately}130fs laser pulses to create and nonintrusively probe the plasmas. Laser prepulse was carefully controlled to suppress preionization and give unambiguous, high-density plasma results. In metals (Al and Au), it is shown analytically that linear and nonlinear inverse Bremsstrahlung absorption, resonance absorption, and vacuum heating explain time-integrated reflectivity at intensities near 10{sup 16}W/cm{sup 2}. In the insulator, SiO{sub 2}, a non-equilibrium plasma reflectivity model using tunneling ionization, Helmholtz equations, and Drude conductivity agrees with time-integrated reflectivity measurements. Moreover, a comparison of ionization and Saha equilibration rates shows that plasma formed by intense, …