A novel and robust scale-invariant WENO scheme for hyperbolic conservation laws

A novel, simple, robust, and effective modification in the nonlinear weights of the scale-invariant WENO operator is proposed that achieves an optimal order of accuracy with smooth function regardless of the critical point (Cp-property), a scale-invariant with an arbitrary scaling of a function (Si-property), an essentially non-oscillatory approximation of a discontinuous function (ENO-property), and, in some cases, a well-balanced WENO finite difference/volume scheme (WB-property) (up to machine rounding error numerically). The classical WENO-JS/Z/D operators do not satisfy the Si-property intrinsically due to a loss of sub-stencils' adaptivity in the WENO reconstruction of a discontinuous function when scaled by a small scaling factor. By introducing the descaling function, an average of the function values in the weights to build the scale-invariant WENO-JSm/Zm/Dm operators, the operators are independent of both the scaling factor and sensitivity parameter. The Si-property and Cp-property of the WENO operators are validated theoretically and numerically in quadruple-precision with small and large scaling factors and sensitivity parameters. The results show that the WENO-JSm/Zm/Dm operators satisfy the Si-property and the WENO-D/Dm operators satisfy the Cp-property. Furthermore, the ENO-property of the WENO-Zm/Dm schemes is illustrated via several one- and two-dimensional shock-tube problems. In solving the Euler equations under gravitational fields, the well-balanced scale-invariant WENO schemes achieve the WB-property intrinsically without imposing the stringent homogenization condition.