Higher-order-accurate upwind schemes for solving the compressible Euler and Navier-Stokes equations

A fifth-order compact upwind TVD scheme and a fourth-order compact MUSCL TVD scheme are proposed for solving the compressible Euler and Navier-Stokes equations. The fundamental form of the present schemes is based on the second(third)-order-accurate upwind scheme. One of the distinctive points using the present MUSCL TVD scheme is the ability to capture the discontinuities, such as slip lines or contact surfaces as well as shocks, more sharply than the existing TVD scheme with a simpler algorithm than the so-called ENO scheme. The algorithms are relatively simple and the formulas are quite compact. They can be applied easily to the existing Euler and Navier-Stokes solvers based on the second(third)-order upwind scheme. Finally, we show some numerical results of steady and unsteady flows, including shocks, weak discontinuities and vortices, and the superiority of the present scheme is confirmed by comparison with the results of the ordinary numerical scheme.