A new fifth order finite difference WENO scheme to improve convergence rate at critical points

In the present work, we construct a new, improved version of fifth-order finite-difference Weighted Essentially Non-Oscillatory (WENO) scheme with less dissipation to approximate solutions for one- and two-dimensional hyperbolic conservation laws and associated problems. The higher-order information presented for the classical WENO scheme is applied to construct a new global smoothness indicator, a composition of first derivatives of local smoothness indicators with their respective interpolation polynomials. The sufficient condition of nonlinear weights is satisfied by using Taylor expansions. We verify the proposed scheme through several benchmark test cases for scalar and system of hyperbolic conservation laws. We observe from the numerical simulations that the designed WENO scheme demonstrates high resolutions, fifth-order accuracy, and robustness at critical points as compared with other modern WENO schemes. • In this paper, we have proposed a new fifth order WENO scheme that provides more accurate results. • We have formulated a new global smoothness measurement by combining all the first derivatives of local smoothness measurements for candidate sub-stencils ( τ N Z = | β . 0 − 2 β . 1 + β . 2 | ) . • By using this new global smoothness measurement, the finite difference WENO scheme provides fifth order accuracy at critical points. • By using this indicator, the computational results have been found more accurate than WENO-JS5, WENO-Z5 and WENO-P5 schemes. • We have also shown some results for kinetic flux splitting criteria. • For two-dimensional cases, we have shown results for Explosion problem, Doble Mach Reflection problem of a strong shock, Configuration 3 and Configuration 12. • We can clearly observe from the figures that the new proposed scheme have provided bigger and better resolutions of vortices than WENO-JS5, WENO-Z5 and WENO-P5 schemes. • We have also observed that the new scheme is showing more number of vortices near the spiral regions than other considered schemes.