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.
The purpose of this paper is to develop a robust and efficient high order fully conservative finite difference scheme for compressible Navier-Stokes equations. The 5th order WENO scheme is used for the inviscid fluxes. A conservative fourth order accuracy finite central differencing scheme is developed for the viscous terms. An improved ε value of 10 −2 is suggested for the WENO smooth factors calculation, which removes the weight oscillation and significantly improves the convergence rate and level. The wall surface is taken as half-point mesh so that the no slip wall boundary condition can be accurately imposed. A 3th order accurate finite difference scheme is given to treat wall boundary condition. The implicit time marching method with unfactored Gauss-Seidel line relaxation is used with the high order scheme to achieve steady state solutions with high convergence rate.
A low diffusion E-CUSP (LDE) scheme is applied with 5th order WENO scheme in this paper. The E-CUSP scheme can capture crisp shock profile and exact contact surface. Several numerical cases are presented to demonstrate the accuracy and robustness for the E-CUSP scheme to be used with the WENO strategy.
SUMMARYA fifth-order accurate multistep weighted essentially non-oscillatory (WENO) scheme is constructed in this paper. Different from the traditional WENO schemes, which are designed to have (2r 1/th order accuracy in the smooth regions directly from r candidate stencils, the new scheme is constructed through (r 1/ weighting steps. In each step, only two neighboring stencils are used to construct the intermediate fluxes (or the final flux), which are only one order higher than the fluxes obtained from the previous step. Henrick's mapping function is used in each step to satisfy the sufficient condition of fifth-order convergence for a fifth-order WENO scheme; hence, the new scheme is fifth-order accurate in smooth regions. The distinctive advantage of the new scheme is that it can improve the accuracy by one order higher than the traditional WENO schemes at transition points (connecting a smooth region and a discontinuity point); and hence, it improves the accuracy in the regions near discontinuities. Numerical examples show that the new scheme is robust and is less dissipative than the traditional fifth-order WENO schemes.
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