We study numerical integration over arbitrary interface and arbitrary domain on block-structured adaptive mesh.Arbitrary interface and arbitrary domain are described by a level set function.We first describe numerical methods on uniform Cartesian grid.Then we extend the methods to block-structured adaptive mesh.Numerical calculations demonstrate that adaptive mesh methods are second-order accurate.Compared with uniform mesh methods, the adaptive mesh methods reduce needs on computer storage significantly.

We analyze approximation and propose a two-level algorithm for finite volume schemes of diffusion equations with structured adaptive mesh refinement. First of all, a typically conservative finite volume scheme was discussed, along with criterion for refining and coarsening interpolation operator. Secondly, non-conforming elements around coarse-fine interface were eliminated by introducing auxiliary triangle elements. A symmetric finite volume element (SFVE) scheme was designed. And further analysis showed the scheme has better approximation. It weakens restrictions. Finally, a two-level algorithm was constructed for SFVE. Theoretical analysis and numerical experiments demonstrate uniform convergence of the algorithm.

Performance of physical-variable based coarsening two-level(PCTL) preconditioner is analyzed for typical linear systems discretizated from two-dimensional(2-D) radiative diffusion equations with photon,electron,ion temperatures(3-T).It reveals that performance of PCTL strongly depends on both coupling of three temperatures and diagonally dominance of three diagonal sub-matrices of coefficient matrix.An adaptive algorithm for sub linear systems in PCTL is proposed.Numerical results show efficiency and robustness of the method.For 37 2-D 3-T linear systems in simulations,PCTL based on the algorithm speeds up 2.5 times compared with classical algebraic multigrid (AMG) preconditioners.

Preconditioners for GMRES method are discussed in solving linear systems discretized from scalar elliptic partial differential equations of second order with jump coeffcient.Based on hierarchical basis,spectral equivalence is established for two kinds of stiffness matrices from quadratic finite element and second order mixed-type finite volume element method,respectively.A preconditioner is proposed by combining equivalence with two-level geometric-algebraic multigrid(GAMG) method which was especially designed for linear systems arising from quadratic finite element discretization.Numerical results confirm correctness of our theoretical analysis.It shows that the preconditioner is quite effcient and robust.

With special control volumes and finite volume element spaces,two symmetry-preserving finite volume element schemes for stationary diffusion problems are established on unstructured quadrilateral grids.Saturated order of error in L²-norm and H¹-norm for discrete solutions under quasi-uniform partition is demonstrated as diffusion coefficient is smooth.Numerical examples verify theoretical results.It shows strong adaptability of the scheme on distorted quadrilateral grids and for diffusion problems with non-smooth coefficient.The second scheme shows super-approximation for flux function at central point of element as grids are orthogonal.

High order weighted essentially non-oscillatory finite difference schemes (FD-WENO) are applied successfully to numerical simulation of gravity-driven high density ratio Rayleigh-Taylor instability problems and laser ablative Rayleigh-Taylor instability problems in two dimensions. It provides important references to numerical study of inertial confinement fusion (ICF) as well as to other high-tech fields. High order FD-WENO schemes are applicable to numerical simulation of ICF inplosion.

A mesh adaptation approach based on Hessian matrix is proposed to solve two-dimensional heat conductione equations with coupled electron,iron and photon temperatures.Three kinds of adaptive mesh and two adaptation methods based on gradient and flux of the photon finite element solution are used. It is shown that the energy conservation error and computation efficiency of the approach are improved.

Two kinds of algebraic multigrid (AMG) algorithms with three-dimensional equal algebraic structures are constructed on the basis of a two-dimensional coarsing technique.The AMG method and the corresponding algebraic multigrid-preconditioned CG method are applied to elliptic boundary value problems with smooth coefficients and anisotropic problems.Numerical results show that the AMG algorithm is efficient and robust.