New third-order weighted essentially non-oscillatory scheme with novel compact nonlinear weights based on the forward-divided differences

The current paper proposes a new third-order WENO (weighted essentially non-oscillatory) scheme, denoted as WENO-D3, which constructs the numerical flux using the convex combination of a second-degree polynomial of a three-point stencil with two linear polynomials of two substencils. WENO-D3 comes with a compact formulation of the numerical flux, new compact nonlinear weights, a new reference smoothness indicator, and a simple smoothness indicator for the three-point stencil. The forward differences approach is employed to reformulate the expressions of the polynomials and smoothness indicators of the new scheme. The smoothness indicator of the three-point stencil is designed using a linear combination of the two-point substencil smoothness indicators and compact linear weights. The linear weights of WENO-D3 can be freely selected with one condition: their sum equals one. A detailed analysis of the WENO-D3 scheme is provided, and numerous one- and two-dimensional benchmark numerical experiments are studied. Thirty-one combinations of linear weights are studied to verify the sensitivity of WENO-D3 to linear weights selection. Compared to the WENO-MZQ3 scheme, the WENO-D3 scheme significantly reduced the computational complexity and cost while providing flexibility in linear weights selection. The results show that the new scheme reduces up to 91% of the WENO-MZQ3 scheme's computational time and provides stable results for a wide range of linear weights. The results also show that the proposed scheme has recovered the optimal order at critical points and can capture and resolve sharp discontinuities without spurious oscillations.