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This paper studies the weights stability and accuracy of the implicit fifth-order weighted essentially nonoscillatory finite difference scheme. It is observed that the weights of the Jiang-Shu weighted essentially nonoscillatory scheme oscillate even for smooth flows. An increased " value of 10 2 is suggested for the weighted essentially nonoscillatory smoothness factors, which removes the weights oscillation and significantly improves the accuracy of the weights and solution convergence. With the improved " value, the weights achieve the optimum value with minimum numerical dissipation in smooth regions and maintain the sensitivity to capture nonoscillatory shock profiles for the transonic flows. The theoretical justification of this treatment is given in the paper. The wall surface boundary condition uses a half-point mesh so that the conservative differencing can be enforced. A third-order accurate finite difference scheme is given to treat wall boundary conditions. The implicit time-marching method with unfactored Gauss-Seidel line relaxation is used with the high-order schemes to achieve a high convergence rate. Several transonic cases are calculated to demonstrate the robustness, efficiency, and accuracy of the methodology. Nomenclature C k = optimal weight IS k = smoothness estimator J = Jacobian of transformation M = Mach number Pr = Prandtl number Pr t = turbulent Prandtl number p = pressure/power used for weighted essentially nonoscillatory scheme q k = heat flux in Cartesian coordinates/third-order polynomial interpolation Re = Reynolds number t = time u, v, w = velocity components in x, y, and z direction x, y, z = Cartesian coordinates = ratio of specific heats U = difference of the conservative variables " = parameter introduced in weighted essentially nonoscillatory scheme = molecular viscosity t = turbulent viscosity , , = generalized coordinates = density ! k = weight Subscripts i, j, k = indices w = wall 1 = freestream Superscripts L, R = left and right sides of the interface n = time level = dimensionless variable

In this work, a new smoothness indicator that measures the local smoothness of a function in a stencil is introduced. The new local smoothness indicator is defined based on the Lagrangian interpolation polynomial and has a more succinct form compared with the classical one proposed by Jiang and Shu [12]. Furthermore, several global smoothness indicators with truncation errors of up to 8th-order are devised. With the new local and global smoothness indicators, the corresponding weighted essentially non-oscillatory (WENO) scheme can present the fifth order convergence in smooth regions, especially at critical points where the first and second derivatives vanish (but the third derivatives are not zero). Also, the use of higher order global smoothness indicators incurs less dissipation near the discontinuities of the solution. Numerical experiments are conducted to demonstrate the performance of the proposed scheme.

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