Accurate numerical algorithms for solving systems of nonlinear hyperbolic equations are considered. The issues of the capturing and the non-diffusive resolution of contact discontinuities were investigated using two different approaches: a kinetic fluctuation splitting scheme and a discontinuity confinement scheme based on an antidiffusion approach. In both approaches cell-vertex fluctuation-splitting methods are used in order to generate a multi-dimensional procedure. The kinetic fluctuation-splitting scheme presented here is a Boltzmann type scheme based on an LDA-scheme discretization on a triangulated Cartesian mesh that uses diagonal adaptive strategy. The LDA scheme developed by Struijs, Deconinck and Roe has the property of being second-order accurate and linear for a scalar advection equation. It is implemented for the Boltzmann equation following the work of Eppard and Grossman and completes the series of multi-dimensional Euler solvers with upwinding applied at the kinetic level. The MKFS-LDA scheme is a cell-vertex scheme. It was obtained by taking the moments of the fluctuation in the distribution function that are calculated according to the LDA fluctuation splitting procedure on a kinetic level. The diagonal-adaptive procedure designed by Eppard and Grossman for MKFS-NDA scheme was applied to eliminate the diagonal dependence. Results show improvement over lower-order N-scheme based solvers. Results for a simple oblique-shock reflection and a shear wave demonstrate that the adaptive procedure and the higher order low diffusion scheme provide sharper resolution than the dimensionally-split kinetic CIR scheme and the first order N-scheme. Moreover, no evidence of oscillations near discontinuities was observed but the order of accuracy is probably lower than the second-order theoretically predicted accuracy on regular meshes. Results for the inviscid reflection of an oblique shock wave and for an oblique shear wave indicate greatly improved resolution over first-order dimensionally-split approach. However, the complexity of the scheme and CPU usage increase were not justified by slight improvement over the first order N-scheme. In the second part a new discontinuity confinement procedure is described. It uses ideas developed by Steinhoff to capture concentrated vorticity layers and short acoustic pulses. The discontinuity confinement method not only captures contact discontinuities over a few grid cells but also safeguards them from numerical dissipation as they evolve with time and over long spatial distances. A one dimensional discontinuity confinement method in terms of flux correction was developed and applied to the Euler solver. The results for the shock tube problem and contact discontinuity propagation were encouraging but the extension to two dimensional problems was not possible in the flux correction framework. A new scheme was developed using a closely related antidiffusion convection and fluctuation splitting ideas used in vortex confinement method by Steinhoff. The new formulation provides a simple and multi-dimensional procedure that can be used with any monotone basic solver. A comparison of the dissipative property of the confinement scheme with a higher-order dimensionally split upwind scheme and several solutions on adaptive unstructured grids demonstrate that the new method has the ability to much more sharply resolve complex regions with contact discontinuities. Moreover, the quality of the solution does not deteriorate over many time iterations or long spatial distances. Solutions for several two-dimensional steady problems are presented to demonstrate the high resolution property of the new scheme. It includes oblique shear layer problem, triple point problem and underexpanded nozzle flow. Underexpanded nozzle flow involves complex interaction of free surface, shock waves and slip lines. Traditional high-order schemes smear the contact surfaces preventing accurate definition of the flow structure inside the jet. However, using a Cartesian grid and the discontinuity confinement procedure the free surface of the jet and slip lines were resolved within 3 grid cells. In the case of an oblique shear layer the new scheme demonstrated no degradation of the initial profile for the discontinuity. Robustness of the confinement scheme depends on the monotone basic solver and also multi-dimensional switch that excludes expansion regions from effect of the confinement. Research still needs to be done in order to develop a switch which can be used on a wide variety of applications. / Ph. D.