A SCALE-INVARIANT HIGH-ORDER WCNS SCHEME

For numerical simulation of high-speed flows, it requires that small-scale structures are resolved with high-fidelity, and discontinuities are stably captured without spurious oscillation. These two aspects put forward almost contradictory requirements for numerical schemes. The widely used high-order schemes can satisfy the two demands required above to some extent. However, they all have advantages and disadvantages compared to each other and no one can be considered perfect. For example, high-order schemes are prone to generating numerical oscillations near discontinuities when a small-scale problem is discretized, such as the Reynolds-stress model. To solve this deficiency, a simple, effective and robust modification is introduced to the seventh-order weight compact nonlinear scheme (WCNS) by making use of the descaling function to formulate a scale-invariant WCNS scheme. The descaling function is devised using an average of the function values and introduced into the nonlinear weights of the WCNS7-JS/Z/D schemes to eliminate the scale dependency. The design idea of the scale-invariant WCNS scheme is to make weights independent of the scale factor and the sensitivity parameter. In addition, the shock-capturing ability of the new scheme performs well even for small-scale problems. The new schemes can achieve an essentially non-oscillatory approximation of a discontinuous function (ENO-property), a scale-invariant property with an arbitrary scale of a function (Si-property), and an optimal order of accuracy with smooth function regardless of the critical point (Cp-property). We derive the seventh-order D-type weights. The one-dimensional linear advection equation is solved to verify that WCNS schemes can achieve the optimal (seventh) order of accuracy. We test a series of one- and two-dimensional numerical experiments governed by Euler equations to demonstrate that the scale-invariant WCNS schemes perform well in the shock-capturing ability. Overall, the scale-invariant WCNS schemes provide a new method for improving WCNS schemes and solving nonlinear problems.