Implementing high-order weighted compact nonlinear scheme on patched grids with a nonlinear interpolation

•A new sixth-order weighted nonlinear interpolation is developed.•The interpolation can avoid interpolation oscillations for shock wave problems.•The interpolation may be useful in CFD on coping patched grids and overset grids.

Weighted Compact Nonlinear Schemes (WCNSs) possess the merits of high-order accuracy, high resolution, and high endurance in low quality grids. In order to implement WCNS on patched grids, high-order interpolations are generally needed for the purpose of ensuring overall high-order accuracy. However, high-order linear interpolations are susceptible to numerical oscillations in the vicinity of discontinuities and large gradient regions. In order to prevent this kind of oscillations, a new weighted interpolation is developed following the idea of Deng and Zhang (J Comput Phys, 2000). The new interpolation contains three fourth-order sub-interpolations which are weighted together according their smoothness. The weights are designed in such a way that in smooth regions they could approach to the optimal weights to achieve sixth-order accuracy, whereas in regions near discontinuities, the weights of the stencils which contain discontinuities are assigned to be nearly zero. The fifth-order WCNS is combined with the new weighted interpolation to solve flow problems on patched grids. Several benchmark problems, including shock waves, vortex, and shock/vortex interaction, are simulated to test the method. The results indicate that the new interpolation has similar performance with the sixth-order Lagrange interpolation in smooth regions, and is superior to the Lagrange interpolation in preventing interpolation oscillations in the vicinity of discontinuities.