Improvement of Stability and Accuracy for Weighted Essentially Nonoscillatory Scheme

Abstract

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

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

“…In their paper [53], e is taken as 10 À6 . In [54], Shen et al suggested to use an optimized e value of 10 À2 in the smoothness estimators to achieve optimal weight in smooth regions in order to minimize dissipation and improve convergence.…”

Low diffusion E-CUSP scheme with implicit high order WENO scheme for preconditioned Navier–Stokes equations

“…In this paper, at boundaries, the conservative variables are first obtained using various required boundary conditions [43], and then the primitive variables used in preconditioning system are calculated from the conservation variables accordingly.…”

“…In Ref. [43], the e value of 10 À2 suggested by Shen et al to suppress the oscillation of IS k and improve the convergence and accuracy is adopted in this paper. The q R is constructed symmetrically as q L about i + 1/2.…”

Comparison of Drag Prediction Using RANS models and DDES for the DLR-F6 Configuration Using High Order Schemes

“…They introduced fourth-order fluxes to overcome this drawback. For transonic flows, Shen et al [14] suggested to use an optimized ε in the smoothness estimators to achieve optimal weight in smooth regions in order to minimize dissipation. A class of higher than 5th order weighted essentially non-oscillatory schemes are designed by Balsara and Shu in [15].…”

“…The implicit time marching method with unfactored Gauss-Seidel line relaxation is used with the 5th order WENO finite difference scheme described in [14] for solving the Navier-Stokes equations. The viscous terms are discretized using the 4th order conservative central differencing suggested by Shen and Zha [17].…”