A Fourth-order MUSCL Finite-Difference Scheme for Solving the Unsteady Compressible Euler Equations

A fourth-order compact MUSCL (Monotone Upstream-centered Scheme for Conservation Laws) TVD (Total Variation Diminishing) scheme is proposed for solving the unsteady compressible Euler Equations. The fundamental form of the present scheme is based on the second (third) -order accurate MUSCL finite-difference scheme. One of the distinctive features of this scheme is the ability to capture discontinuities such as slip lines or contact surfaces as well as shocks more sharply than the existing TVD schemes in spite of the use of a simpler algorithm than that of the so-called ENO scheme. Therefore, this scheme can be easily applied to any ordinary numerical solvers based on the second (third)-order MUSCL scheme. We also adopt this scheme into the existing Euler solver developed by the authors which can simulate unsteady inviscid flows accurately by means of the Newton iteration and the Crank-Nicholson method. In order to verify the reliability of the present scheme, an unsteady inviscid supersonic flow having oblique shocks and a slip line is computed. The results show that the slip line as well as oblique shocks can be captured completely.