(untitled) (Advances in Applied Mathematics and Mechanics, 2023)
In this article, a robust, effective, and scale-invariant weighted compact nonlinear scheme (WCNS) is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes. The new scheme achieves 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). The classical WCNS-Z/D schemes do not satisfy Si-property intrinsically, which is caused by a loss of sub-stencils’ adaptivity in the nonlinear interpolation of a discontinuous function when scaled by a small scale factor. A new nonlinear weight is devised by using an average of the function values and the descaling function, providing the new WCNS schemes (WCNS-Zm/Dm) with many attractive properties. The ENO-property, Si-property and Cp-property of the new WCNS schemes are validated numerically. Results show that the WCNS-Zm/Dm schemes satisfy the ENO-property and Si-property, while only the WCNS-Dm scheme satisfies the Cp-property. In addition, the Gaussian wave problem is solved by using successively refined grids to verify that the optimal order of accuracy of the new schemes can be achieved. Several one-dimensional shock tube problems, and two-dimensional double Mach reflection (DMR) problem and the Riemann IVP problem are simulated to illustrate the ENO-property and Si-property of the scale-invariant WCNS-Zm/Dm schemes.