The multi-flow tube (MFT) method is mainly designed for two-dimensional (2-D) radiation hydrodynamic problems with multi-material, whose computational mesh is Lagrangian-Eulerian hybrid. Quality of the MFT quadrilateral mesh is usually higher than that of a general quadrilateral mesh. We consider a so-called nine-point diffusion scheme on this kind of meshes, in which the method of calculating vertex unknowns is improved by taking advantage of mesh features. Several methods for eliminating vertex unknowns are proposed by using harmonic averaging points or a gradient reconstruction method. Numerical experiments show that our methods obtain almost second order accuracy on MFT meshes.

Based on a second-order accurate linear scheme, a normal flux is reconstructed to obtain a nonlinear finite volume scheme with a two-point flux discrete stencil on tetrahedral meshes. It is suitable for discontinuous and anisotropic diffusion coefficient problems, and can be generalized to general polyhedral meshes. It is unnecessary to assume that auxiliary unknowns are non-negative, and avoids artificial processing of "setting negative to be zero" in calculating auxiliary unknowns. Moreover, it is proved that the linearized scheme at each nonlinear iteration step satisfies strong positivity-preserving, i.e., as the source term and boundary condition are non-negative, non-zero solution of the scheme is strictly greater than zero. Numerical tests verify that the scheme has second-order accuracy and is strong positivity-preserving.

In order to improve robustness of multi-fluid-channel scheme on average of volume (MFCAV) and overcome its artificial intervention in actual application, a new HLLCM scheme was designed on a moving mesh, which restrains non-physical mesh tangling without artificial intervention. Numerical results show that the HLLCM scheme maintains one-dimensional spherical symmetry on equal-angle-zoned grids which has better numerical effects in keeping mesh quality and energy conservation in complex applications than MFCAV scheme.

In practical applications, equation of states (EOS) consists of several piecewise smooth surfaces, which leads to discontinuity at interface. As a traditional nonlinear iterative algorithm is applied to an energy equation with discontinuous EOS, it may lead to slow convergence and unphysical solutions. To overcome the difficulties, a nonlinear problem is designed, and a nonlinear iterative algorithm is proposed to solve the problem. The algorithm is fit for energy equations with discontinuous EOS of piecewise smooth functions. A parameter of energy change is defined in the algorithm so that it is unnecessary to know discontinuity position in advance. The algorithm calculates precisely net gain or leakage of energy, which can be used to assess influence of discontinuity in EOS. Typical numerical experiments verify that the algorithm converges stably, and gives physical solutions.

We discuss deterministic numerical methods such as discrete ordinates methods,spherical harmonic dimensional and iteration acceleration method,for particle (neutron and radiation) transport equations in one-dimensional spheric geometry and twodimesional cylindrical geometry. Recent advances on numerical methods of transport problems are briefly described,including adaptive time step discrete scheme,iterative methods for eigenvalue problems,modified subcell balance methods,simplified spherical harmonic methods,simulation methods for coupling of diffusion and transport,parallel discrete scheme with interface prediction and correction,grey transport synthetic acceleration method,etc. Based on the analysis of difficulties in numerical methods for transport equations,suggestions for future work are proposed.

We give a convergence analysis on splitting iterative (SI) algorithm of multi-group radiation diffusion equations. Spectral radii of iterative matrix is shown. Numerical computation and analysis on spectral radii formulae reveal a relation between convergence rate and radiation coefficients. Numerical results confirm theoretical results, and give applicable conditions of the algorithm.

We propose a method of time step for predicator-corrector scheme of Lagrangian method on stagger grids.It is different from CFL time step based on tradional stable theory.The method considers nonlinear effect of original partial differential equations,and gives an adaptive time step based on physical quantities positivity preserving.Numerical results confirm validity of the method.

For discontinuous finite element method of time-dependent neutron transport equations in 2D cylindrical geometry,a parallel algorithm with interface prediction and correction is designed.Numerical experiments show that the parallel algorithm with explicit prediction and implicit correction has good precision,parallelism and simplicity.Compared with parallel algorithm based on implicit scheme,the new parallel algorithm has higher parallel efficiency with accuracy-preserving.Especially,it achieves super-linear speedup on hundreds of processors for large-scale problems.

Nine.point scheme for diffusion equation on distorted quadrilateral is discussed.Based on a classical nine。point scheme, and with continuity of diffusion flux, a formula for designed for computing Vertex unknowns. Numerical experiments show that the scheme is robust,and the accuracy is higher than that of other weighted methods for diffusion problem with continuous or discontinuous coefficient on distorted meshes.

An explicit method is proposed to initiate three-layer difference scheme for two-dimensional heat problems.Stability and convergence theorem is shown with a three-layer domain decomposition parallel difference algorithm with inner boundary prediction-correction and explicit initialization.Parallel numerical experiments show that the method is numerical stable,and is more convenient in program realization compared with usual implicit initialization method.It reduces numerical errors greatly compared with existing perturbation methods.

In numerical simulation of radiation hydrodynamics problems,diffusion computation costs most part of simulation time.It is urgent to construct parallel schemes which are conservative,of high precision and unconditionally stable.The aim of the paper is to devise conservative parallel schemes based on methods of "prediction and/or correction" for nonlinear diffusion equations.Numerical results are presented to examine performance of conservative parallel schemes,such as accuracy,stability and parallel speedup.

Radiative transfer in fluid flow of radiation hydrodynamics is studied.Kinetic laws under radiation condition are investigated.Practical radiation hydrodynamics process is complicated,and numerical simulation is one of primary research means.Splitting methods are often used in numerical simulation,in which fluid motion and radiation are computed separately.We discuss computational problems in radiation diffusion calculations.Diffusion schemes and nonlinear iterative methods on severely distorted meshes are studied.A brief introduction on research progress is given.

A linear discontinuous spatial finite element scheme for time-dependent particle transport equation is studied.Numerical precision is considered through error norms.Numerical precision of linear discontinues finite element method on edge of each cell is higher than those of exponential method and diamond difference method.It shows that linear discontinuous finite element method is more accurate and differential curve on time about flux is more smooth than that of exponential method and diamond difference.

A discrete tangential flux on grid edge in nine-point schemes for solving diffusion equations on arbitrary quadrilateral meshes is derived. The discrete tangential flux is represented as a weighted combination of difference of center values with adaptive weighted coefficients, which depend on the thermal conductivity and geometric parameters of distorted meshes.

A structured grid generation method for domains with complicated boundary is discussed.Based on the Winslow method with variational form,and combined with grid untangling and area averaging technologies,a discrete functional is designed. The minimization of the discrete functional is solved by an optimization algorithm,and good grids are generated.Numerical experiments show that the method is robust and generates grids with good geometric qualities on complicated domains.This method inherits advantages of the Winslow method and overcomes some faults.

Construction of difference schemes for parabolic equations on distorted meshes with large aspect ratio is discussed.The limitation of 9-point scheme is provided.An assistant mesh difference method is proposed for distorted meshes with large aspect ratio to improve precision and efficiency.Based on the conservation law,applying appropriate rezone methods for meshes we modify the 9-point method to reduce computation error due to poor regularity of the meshes and low precision due to the approximation to vertices values by an average of neighboring cell values.The nonlinear system obtained is a different system from the 9-point scheme.Its solution approximates to the solution with the original meshes.The designed scheme is implemented easily and adapts well to distorted meshes.Numerical experiments show good precision and stability.

Combining reference Jacobian method(RJM) with advancing-front method,we present a strategy named advancing reference Jacobian method(ARJM).It advances the optimization process step by step from one part of the computational region boundary to the remaining parts.In each step,two neighboring rows(or columns) are taken as the boundaries of the sub-region and the rear row(column) nodes are the optimized ones and the middle row(or column) is optimized by RJM. Analyses and numerical experiments show that the ARJM is much faster than RJM.The geometric qualities of rezoned grids by ARJM are equal to or even better than those by RJM.The rezoned grids obtained by ARJM are closer to Lagrangian grids than those by RJM.