INVESTIGATION OF THE TIME EFFICIENCY OF THE SEVENTH-ORDER WENO-S SCHEME

The WENO-S scheme is a class of weighted essentially non-oscillatory schemes suitable for numerical simulations of problems with discontinuities. The smoothness indicator of this kind of scheme is constant for single-frequency waves, which makes this kind of scheme have exactly the same approximate dispersion relationship with its linear base scheme, and thus has an excellent ability to simulate small-scale waves. Time efficiency is crucial for numerical methods. For a WENO-S scheme, the formula of the smoothness indicator on each sub-stencil has the same formula except for different subscripts. Then some smoothness indicators are the same when calculating adjacent numerical fluxes of linear convection equations. So, a method is proposed to remove redundant computations of smoothness indicators. The premise of this approach is that the quantity used for reconstruction or interpolation on a grid line can be represented as a sequence. According to this requirement, the feasibility and application requirements for several different physical problems are analyzed. The seventh-order WENO-S scheme is employed to illustrate the advantages of the WENO-S schemes, including good properties near extreme points, good stability near discontinuities, and outstanding spectral properties. Then the method of eliminating the computation of the redundant smoothness indicators is introduced. In numerical computation, all smoothness indicators in a grid line are calculated and stored in advance. With this approach, the count of the smoothness indicator calculation is about 1/4 of the original one for the seventh-order WENO-S scheme when there are many grid points. Numerical examples include one-dimensional advection, spherical wave propagation, two-dimensional rotation, small disturbance propagation, and one- and two- dimensional inviscid flow problems. The numerical results show that this scheme can capture a variety of flow structures well and have good time efficiency. Furthermore, the proposed method reduces the computational time by about 20%.