Abstract—In this work, first part of this study, the high resolution numerical schemes of Lax and Wendroff, of Yee, Warming and Harten, of Yee, and of Harten and Osher are applied to the solution of the Euler and Navier-Stokes equations in two- dimensions. With the exception of the Lax and Wendroff and of the Yee schemes, which are symmetrical ones, all others are flux difference splitting algorithms. All schemes are second order accurate in space and first order accurate in time. The Euler and Navier-Stokes equations, written in a conservative and integral form, are solved, according to a finite volume and structured formulations. A spatially variable time step procedure is employed aiming to accelerate the convergence of the numerical schemes to the steady state condition. It has proved excellent gains in terms of convergence acceleration as reported by Maciel. The physical problems of the supersonic shock reflection at the wall and the supersonic flow along a compression corner are solved, in the inviscid case. For the viscous case, the supersonic flow along a compression corner is solved. In the inviscid case, an implicit formulation is employed to marching in time, whereas in the viscous case, a time splitting approach is used. The results have demonstrated that the Yee, Warming and Harten algorithm has presented the best solution in the inviscid shock reflection problem; the Harten and Osher algorithm, in its ENO version, and the Lax and Wendroff TVD algorithm, in its Van Leer variant, have yielded the best solutions in the inviscid compression corner problem; and the Lax and Wendroff TVD algorithm, in its Minmod1 variant, has presented the best solution in the viscous compression corner problem. Keywords—Lax and Wendroff algorithm; Yee, Warming and Harten algorithm; Yee algorithm; Harten and Osher algorithm; TVD and ENO flux splitting, Euler and Navier-Stokes equations, Finite volume, Two-dimensions. Edisson S. G. Maciel works as a post-doctorate researcher at ITA (Aeronautical Technological Institute), Aeronautical Engineering Division – Praça Marechal do Ar Eduardo Gomes, 50 – Vila das Acácias – São José dos Campos – SP – Brazil – 12228-900 (corresponding author, phone number: +55 012 99165-3565; e-mail: edisavio@edissonsavio.eng.br). Eduardo M. Ferreira teaches at UFGD (Federal University of Great Dourados), Energy Department – Rodovia Dourados – Itahum, km 12 – Caixa Postal 533 – Cidade Universitária – Dourados – MS – Brazil – 79804- 970 (e-mail: eduardomanfredini@ufgd.edu.br) Amilcar P. Pimenta teaches at ITA (Aeronautical Technological Institute), Aeronautical Engineering Division – Praça Marechal do Ar Eduardo Gomes, 50 – Vila das Acácias – São José dos Campos – SP – Brazil – 12228-900 (e- mail: amilcar@ita.br) Nikos E. Mastorakis is with WSEAS (World Scientific and Engineering Academy and Society), A. I. Theologou 17-23, 15773 Zografou, Athens, Greece, E-mail: mastor@wseas.org as well as with the Technical University of Sofia, Industrial Engineering Department, Sofia, 1000, Bulgaria mailto:mastor@tu-sofia.bg I. INTRODUCTION ONVENTIONAL shock capturing schemes for the solution of nonlinear hyperbolic conservation laws is linear and L2- stable (stable in the L2-norm) when considered in the constant coefficient case ([1]). There are three major difficulties in using such schemes to compute discontinuous solutions of a nonlinear system, such as the compressible Euler equations: (i) Schemes that are second (or higher) order accurate may produce oscillations wherever the solution is not smooth; (ii) Nonlinear instabilities may develop in spite of the L2- stability in the constant coefficient case; (iii) The scheme may select a nonphysical solution. It is well known that monotone conservative difference schemes always converge and that their limit is the physical weak solution satisfying an entropy inequality. Thus monotone schemes are guaranteed not to have difficulties (ii) and (iii). However, monotone schemes are only first order accurate. Consequently, they produce rather crude approximations whenever the solution varies strongly in space or time. When using a second (or higher) order accurate scheme, some of these difficulties can be overcome by adding a hefty amount of numerical dissipation to the scheme. Unfortunately, this process brings about an irretrievable loss of information that exhibits itself in degraded accuracy and smeared discontinuities. Thus, a typical complaint about conventional schemes which are developed under the guidelines of linear theory is that they are not robust and/or not accurate enough. To overcome the difficulties, a new class of schemes was considered that is more appropriate for the computation of weak solutions (i.e., solutions with shocks and contact discontinuities) of nonlinear hyperbolic conservation laws. These schemes are required (a) to be total variation diminishing in the nonlinear scalar case and the constant coefficient system case ([2-3]) and (b) to be consistent with the conservation law and an entropy inequality ([4-5]). The first property guarantees that the scheme does not generate spurious oscillations. Schemes with this property are referred in the literature as total variation diminishing (TVD) schemes (or total variation non-increasing, TVNI, [3]). The latter property guarantees that the weak solutions are physical ones. Schemes in this class are guaranteed to avoid difficulties (i)-(iii) mentioned above. [6] has proposed a very enlightening generalized formulation of TVD [7] schemes. Roe’s result, in turn, is a TVD and ENO Applications to Supersonic Flows in 2D – Initial Study Edisson S. G. Maciel, Eduardo M. Ferreira, Amilcar P. Pimenta, and Nikos E. Mastorakis C INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8, 2014 ISSN: 1998-0140 441