Minimizing errors from linear and nonlinear weights of WENO scheme for broadband applications with shock waves

Abstract

Improvements in the numerical algorithm for the dynamics of flows that involve discontinuities and broadband fluctuations simultaneously are proposed. These two flow features suggest numerical strategies of a paradoxical nature because the discontinuities demand dissipation, and the small-scale smooth features require the opposite. There may be several ways to approach such a complicated issue, but the natural choice is a numerical technique that can adjust adaptively with flow regimes. The weighted essentially non-oscillatory (WENO) scheme may be this choice. However, there are two sources of dissipation associated with the WENO procedure: the upwind optimal stencil and the nonlinear adaption mechanism. The current work suggests a robust and comprehensive treatment for the minimization of dissipation error from these two sources. The optimization technique, which is guided by restriction on the linear optimal weights derived from stability and consistency requirement, is used to delay the dissipation of the upwind optimal stencil to those wavenumbers for which the dispersion error is large. The parallel advantage of this technique is the improvement of the dispersion property. Nevertheless, optimization decreases the formal order of accuracy of the optimal stencil from fifth order to third order. This loss of accuracy is derived by Taylor series expansion. Using Taylor-series expansion and WENO procedure, the third-order accuracy is verified in the smooth region, except at the critical point of order two, where the order of accuracy reduces to at least second order. This possible loss of accuracy at the second-order critical point is restored in an attempt to reduce the dissipation induced by the nonlinear adaptive weights. Modification of the nonlinear weights to reduce the dissipation is introduced by redefining them with an additional smoothness indicator. Other suggestions to minimize the dissipation of the nonlinear weights are also reviewed. The numerical approximation of the spatial derivative is performed by means of a conservative and consistent finite difference method based on monotone local Lax-Friedrichs Riemann solver. The resulting scheme is then integrated by the optimal third-order TVD Runge-Kutta method to ensure the nonlinear stability of the overall numerical method. A variety of benchmark problems, ranging from non-broadband to broadband, are solved using the proposed schemes and compared with the existing ones. Most test problems are validated against exact or reference data. The numerical results with bandwidth optimization and modification of the nonlinear weights are consistently superior.