A third-order weighted essentially non-oscillatory (WENO) scheme is developed for hyperbolic conservation laws on unstructured quadrilateral meshes. As starting point of WENO reconstruction, a general stencil is proposed for any local topology on quadrilateral meshes. With selected stencil, a unified linear scheme was constructed. However, very large weights and non-negative may appear, which leads the scheme unstable even for smooth flows. An optimization approach is given to deal with very large linear weights on unstructured meshes. Splitting technique is considered to deal with negative weights obtained by optimization approach. Non-linear weight with a new smooth indicator is proposed as well. With optimization approach for very large weights and splitting technique for negative weights, the current scheme becomes more robust. Numerical tests are presented to validate accuracy. Expected convergence rate of accuracy is obtained. And absolute value of error is not affected by mesh quality. Numerical results for flow with strong discontinuities are presented to validate robustness of the WENO scheme.

We develop a conservative piece-wise parabolic advecting remapping method(PPRM).The first part of it is an alternate sweeping average method(ASAM) for improving symmetrization of advecting remapping method.The second part is a piece-wise parabolic distributing function for improving order.We use one-and two-dimesional examples to test order and symmetrization of PPRM.

A difference scheme for diffusion equation on Voronoi meshes is constructed using finite volume method.The diffusion discretization scheme is simpler on Voronoi meshes than on quadrilateral meshes.It introduce no cell-node unknown and improves accuracy of the discrete calculation of cell-edge flux as well as accuracy of difference scheme.The diffusion calculation on Voronoi meshes can be coupled with cell-centered hydrodynamics calculations.Computation examples demonstrate that the accuracy on Voronoi meshes is higher than that on quadrilateral meshes and Voronoi meshes adapt well to distortion.

A Godunov-type ALE (Arbitrary Lagrangian-Eulerian) method based on finite volume technique is developed to simulate compressible multimaterial flows with large deformation.The method adapts well to Lagrangian,ALE or Eulerian formulations due to its arbitrary moving speed of meshes.Numerical features are studied in the formulations.Numerical traits of several Godunov-type schemes are studied.Resolutions of scheme for shock and contact are compared.

Variational methods are used in a unified coordinate system.The mesh spacing, smoothness,orthogonality and regularity of grids are considered to obtain an elliptic equation for h.Typical examples demonstrate that it is possible to use variational methods in a unified coordinate system,and the distribution of h satisfies different boundary conditions.