A series of shock capturing schemes based on nonuniform nonlinear weighted interpolation on nonuniform points are developed for conservation laws. Smoothness indicator and discrete conservation laws are discussed. To make fair comparisons between different types of schemes, the properties of eigenvalues of spatial discretization matrices are proved. And the proposed schemes are compared with Weighted Compact Nonlinear Schemes (WCNS) and Flux Reconstruction or Correction Procedure via Reconstruction (FR/CPR) in dispersion, dissipation properties and numerical accuracy. Then, the proposed shock capturing schemes are used as subcell limiters for high-order FR/CPR and the hybrid scheme has superiority in data transformation and satisfying discrete conservation laws. Accuracy, discrete conservation laws and shock capturing properties are tested. Numerical results in one and two dimensions are provided to illustrate that the proposed schemes have good properties in shock capturing and can be applied as subcell limiters for FR/CPR.
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