Based on physical flux Kershaw scheme is extended to multi-block glids and algorithms for degenerative connection or strong connection unstructured points are shown.Flux continuity condition is met in the extended Kershaw scheme on multi-block girds which keeps energy conservation.Three numerical examples show good agreement with exact solutions. It validates accuracy and robustness of the extended Kershaw scheme on multi-block grids.

An integrated gradient total symmetry-preserving(IGTSP) scheme for momentum equations of Lagrangian hydrodynamics in cylindrical coordinates is presented.The scheme inherits advantages of both IGT and IGA schemes,that is,the scheme not only overcomes nonsymmetry of the IGT scheme but also has a momentum conservation error of order O(h),one order higher than that of IGA scheme.A number of numerical examples are tested to demonstrate advantages of IGTSP scheme.

The element-free Galerkin method (EFGM) is improved with partition of unity quadrature (PUQ).Partition of unity quadrature is shown strictly with finite covering and partition of unity.Using Shepard functions as partition of unity functions,we obtain good results.The EFGM with PUQ is a true meshless method. Computational results of EFGM with partition of unity quadrature are more accurate than those of the traditional EFGM with background quadrature.