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.