Moving Mesh Methods For Solving Conservation Law And Radiation Diffusion Systems

In this thesis,we study the moving mesh method and its application to hyperbolic conservation laws and radiation diffusion equations.In the first part of our work,we shall investigate the combination of the moving mesh method with the WENO scheme(Weighted Essential Non-oscillation scheme).Although the WENO scheme and the moving mesh method each have been successfully applied to many application problems,there is little research work to combine them.This is mainly due to the following four reasons:First,the finite difference WENO scheme can only be used on a uniform or curvilinear grid;Second,when used for mesh adaptation,WENO requires that the mesh be smooth in both space and time;Third,the convergence order of WENO will drop dramatically in regions where the mesh variation is large;Finally,the Jacobian matrix of the coordinate transformation associated with the mesh adaptation should have the same approximation order as the finite difference WENO scheme.For these reasons,it is nontrivial to combine the finite difference WENO scheme with the moving mesh method.The main idea of the WENO method is to use the linear combination of low order numerical fluxes to obtain a high order numerical flux,the nonlinear weights applied to each stencil is the key to the success of the WENO method.The WENO method has the advantages of being uniformly high order,resulting in a much sharper transition near shock,being robust,having no artificial parameter to tune,and performing well for problems with strong shock combined with complex smooth solution regions.It is especially suitable for convection-dominated hyperbolic conservation law problems.So our main goal is to combine the advantages of WENO and moving mesh schemes to reach even higher computational efficiency.In this research work,we propose three strategies:First of all,we propose to use the moving least squares method to smooth the mesh to satisfy the mesh smoothness requirement of the WENO method;Second,we propose to employ the GCL(Geometry Conservation Law)method to compute the Jacobian matrix with the WENO scheme so that the derivatives can be approximated at the same order as the WENO method;Finally,we put forward a new type of locally mesh control strategy to overcome the instability caused by large mesh movement.As the results of these effort,we developed a stable moving mesh WENO method and successfully apply it to one-dimensional hy-perbolic conservation law problems.Our numerical examples show that the moving mesh WENO method has the advantages of both the moving mesh and WENO meth-ods.It uses much less mesh points than a uniform mesh method to achieve the same accuracy.It also leads to much sharper discontinuous transitions than a uniform mesh method.Numerical examples also show that the numerical error of the moving mesh WENO method is only about an half of that of a uniform mesh method.The second part of this work is to study the application of the moving mesh method to radiation diffusion equations.Radiation diffusion equations are the gov-erning equation of radiation hydrodynamics.Basically,they are one type of parabolic equations with variable,nonlinear diffusion coefficients,Their main feature is that they have highly nonlinear and/or discontinuous diffusion coefficients.Radiation diffusion equations can be used to model many physical or astronomical phenomena such ans ICF(Inertial Confined Fuse),black-body radiation,and stellar pulsations.Research in this field is of great importance.The main difficulties include the following:First,the highly nonlinear diffusion coefficients cause many iterative methods to fail to converge;Second,large deformation in mesh elements and discontinuous diffusion coefficients make the moving mesh method extremely challenging to apply;Finally,in two or three dimensions,the CPU time cost is huge.For all of these reasons,we employ three strategies to overcome them:We employ an adaptation functional(for the coordinate transformation or adaptive mesh generation)based on equidistribution principle and alignment condition and taking a full consideration of the shape,size and orientation of mesh elements[31];Then we use the well-known "freezing coefficient" method to linearize the diffusion coefficient.This can apparently increase the convergence speed;Finally,we use the so-called cut-off method[49]to ensure the positivity of numerical solution which is crucial to the continuation of the computation.With these strategies,we apply the moving mesh method to multi-material,multiple spot concentration sit-uations of two-dimensional radiation diffusion models.Our numerical examples show that the moving mesh method uses much less mesh points compared to a uniform mesh method to reach the same error level.In order to further enhance the computational efficiency,we propose to use a two-level moving mesh method which employs a coarse mesh for mesh movement while solving the physical equations on a much finer mesh.The numerical examples show that the CPU time cost can be greatly decreased through using the two-level moving mesh method.For the one-dimensional case,our numerical experiments show that our moving mesh scheme uses only an half of the number of uniform mesh points when reaching the same numerical accuracy.