Yiqing Shen

Improvement of Stability and Accuracy for Weighted Essentially Nonoscillatory Scheme

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 e 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 e 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.

Calculation of transonic flows using WENO method with a low diffusion E-CUSP upwind scheme

A low difiusion E-CUSP (LDE) scheme is applied with 5th order WENO scheme in this paper. The E-CUSP scheme can capture crisp shock proflle 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.

Large eddy simulation using a new set of sixth order schemes for compressible viscous terms

A set of conservative sixth order central differencing schemes is suggested for compressible flows with variable viscosity coefficient. This new set of central differencing schemes has the stencil width matching that of the seventh order weighted essentially non-oscillatory scheme (WENO). This feature is important to maintain the compactness of the seventh order WENO scheme and facilitate boundary condition treatment. As an application example, a large eddy simulation (LES) is conducted for a cylinder flow using the seventh order WENO scheme for the convective terms and the new set of sixth order central differencing scheme for the viscous terms. The results are compared with those from other research groups and those obtained using the fifth order WENO scheme and fourth order central differencing.

A new smoothness indicator for improving the weighted essentially non-oscillatory scheme

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

Generalized finite compact difference scheme for shock/complex flowfield interaction

A class of generalized high order finite compact difference schemes is proposed for shock/vortex, shock/boundary layer interaction problems. The finite compact difference scheme takes the region between two shocks as a compact stencil. The high order WENO fluxes on shock stencils are used as the internal boundary fluxes for the compact scheme. A lemma based on the property of smoothness estimators on a 5-points stencil is given to detect the shock position. There is no free parameter introduced to switch the compact scheme and the WENO scheme. Some numerical experiments are given and they demonstrate that the present scheme has low dissipation due to the compact central differencing scheme used in the smooth regions.

High-resolution finite compact difference schemes for hyperbolic conservation laws

A finite compact (FC) difference scheme requiring only bi-diagonal matrix inversion is proposed by using the known high-resolution flux. Introducing TVD or ENO limiters in the numerical flux, several high-resolution FC-schemes of hyperbolic conservation law are developed, including the FC-TVD, third-order FC-ENO and fifth-order FC-ENO schemes. Boundary conditions formulated need only one unknown variable for third-order FC-ENO scheme and two unknown variables for fifth-order FC-ENO scheme. Numerical test results of the proposed FC-scheme were compared with traditional TVD, ENO and WENO schemes to demonstrate its high-order accuracy and high-resolution.

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

An E-CUSP (energy-convective upwind and split pressure) scheme is developed to solve the equations of magnetohydrodynamics. Fifth order WENO reconstructions are employed to calculate the fluxes in order to achieve high order spacial accuracy. A characteristic speed of sound by averaging the fast wave speed and the acoustic speed of sound is suggested to evaluate the Mach number, which will yield robust and accurate solutions. The numerical experiments have demonstrated the accuracy and the capability of the new scheme to capture complex interactions of multiple shocks and vortices.

A Robust Seventh-order WENO Scheme and Its Applications

This paper has analyzed the seventh-order WENO scheme of Balsara and Shu[1] and designed the improved weights for a new seventh-order WENO scheme. The new scheme can not only keep the essentially non-oscillatory property, but can also obtain higher accuracy in smooth regions. Some detailed numerical results are present.

Detached-Eddy Simulations of a Circular Cylinder Using a Low Diffusion E-CUSP and High-Order WENO Scheme

A detached-eddy simulation (DES) of a cylinder turbulent ∞ow using a newly developed low difiusion E-CUSP (LDE) scheme with flfth-order WENO scheme is conducted. A conservative fourth-order accuracy flnite central difierencing scheme is used for the viscous terms. The computed results for the baseline grid agree well with the experiment. The solution sensitivity to spanwise length is investigated. The mesh reflnement is performed. The results show that the mesh reflnement has a signiflcant efiect on the DES simulation.

Comparison Study of Implicit Gauss-Seidel Line Iteration Method for Transonic Flows

This paper studies the sweep direction efiect on the convergence rate and CPU time of the implicit unfactored Gauss-Seidel line relaxation (GSLR) method for compressible ∞ows. The line Gauss-Seidel iteration is also compared with the LU-SGS (lower-upper symmetric Gauss-Seidel) method. A modifled LU-SGS method, namely LU-GSLR, is studied. The LU-GSLR use the unfactored GSLR method with the simple matrix of the LU-SGS. The numerical experiments indicate that for the external ∞ows, the line Gauss-Seidel relaxation methods with sweeping in all directions achieves the optimum convergence rate and CPU e‐ciency. For an inviscid transonic internal ∞ow, the best convergence rate is obtained with sweeping in streamwise direction only. Within each time step, one sweep (a forward sweep plus a backward sweep) per time step is su‐cient. For the three implicit methods, GSLR, LU-SGS, and LU-GSLR, the GSLR is the most e‐cient method when the Roe scheme is used. The LU-GSLR method is a feasible method and can achieve better e‐ciency than the LU-SGS for some cases.