A Study On The Second Order Schemes Of Positivity And Extremum Principle Preserving For High Dimensional Diffusion Equation

Diffusion equations are widely used in industrial manufacturing,reservoir simulation,as-trophysics,plasma physics and other fields.Therefore,it is very important to design efficient and accurate numerical schemes to solve this kind of equations.In the design of numerical schemes,due to the mesh distortion,the anisotropy and discontinuity of diffusion coefficient and other factors,it is still an important challenge to establish numerical schemes preserving physical properties on general meshes.This thesis mainly focuses on this topic,and include five parts:(1)Presenting the proof of non-negativity of vertex unknowns and corresponding Anderson acceleration algorithm,in the study of a vertex unknown eliminating method in the designing of cell-centered positivity-preserving finite volume scheme for diffusion equation on tetrahedral meshes;(2)Positivity-preserving finite volume scheme for diffusion equation on tetrahedral meshes;(3)Extremum principle-preserving finite volume scheme for diffusion e-quation on tetrahedral meshes;(4)Linearized positivity-preserving finite volume scheme for d-iffusion equation on quadrilateral meshes;(5)Property analysis and efficient iterative algorithms of the second-order time accurate finite difference scheme for nonlinear diffusion equation.In the first part,for a vertex interpolation method in the designing of positivity-preserving finite volume schemes for three dimensional diffusion problems[1],i.e.,vertex-moving method,we prove theoretically that the vertex values obtained by this method is non-negative.We also design an iterative acceleration algorithm-Anderson acceleration method based on vertex values,which is suitable for our scheme,and acquire obvious acceleration effect.It overcomes the problem of large amount of calculation due to the complex geometry of three dimensional problems.In the second part,we construct a strong positivity-preserving finite volume scheme for d-iffusion equation on tetrahedral meshes.Different from the existing positive preserving schemes,the new scheme is based on a second-order accurate linear scheme.By nonlinear reconstruc-tion of normal fluxes,the cell-centered positivity preserving finite volume scheme is obtained.One of the main features of the scheme is that it is no longer necessary to assume that the auxiliary unknown is nonnegative.Moreover,it is proved that the Picard linearized scheme at each nonlinear iteration step satisfies strong positivity-preserving,i.e.,when the source term and boundary condition are non-negative,the 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.Then,we apply this method to the discretization of the diffusion term in three-dimensional convection-diffusion equation.A new gradient reconstruction method is used for convective term.The nonlinear coefficient obtained is nonnegative and does not need to be modified.The resulting scheme is conservative and has a local stencil.Moreover,we prove the existence and strong positivity-preserving of the solution for the nonlinear scheme.Numerical results show that our scheme can obtain almost second-order accuracy for the solution,and higher than first-order accuracy for the flux.In the third part,we introduce a new nonlinear finite volume scheme preserving maximum principle for diffusion equations on tetrahedral meshes.By reconstructing the discrete normal flux of an existing second-order linear scheme,we can obtain a discrete flux with local maxi-mum principle structure.In this process,the requirement that the auxiliary unknowns should be expressed as a convex combination of primary unknowns is avoided,which greatly reduces the difficulty of calculating the auxiliary unknown.The resulting scheme is cell-centered and conservative.Numerical examples show that the new scheme has second-order accuracy and satisfies discrete maximum principle.In the fourth part,we construct a novel linearized monotone scheme for diffusion problems on general quadrilateral meshes.It is a combination of linear scheme and nonlinear monotone scheme,and consists of two steps.Firstly,a second-order accurate linear scheme is used to obtain an approximate solution.Secondly,a nonlinear monotone scheme is used to solve the diffusion equation,where the unknowns in the nonlinear coefficients of the nonlinear scheme are taken as the approximate solution of the linear scheme above,i.e.,a linearized monotone scheme is obtained.Consequently,the combination scheme does not require nonlinear iterations when solving linear diffusion problems.We also analyze some properties satisfied by the combination scheme,such as conservation,stability,monotonicity and convergence.In the last part,we study a finite difference scheme with second-order time evolution for nonlinear diffusion equations.Compared with the classical Crank-Nicolson scheme and the second-order backward Euler scheme,this scheme is a two-layer coupled discretization(TL-CD)scheme,which allows a large time step without numerical oscillation.By introducing a new induction reasoning technique,we prove the existence,uniqueness and unconditional sta-bility of the TLCD solution,and its second-order convergence in both space and time to the solution of the original PDE.We also give the corresponding Picard,Picard-Newton(PN)and derivative-free Picard-Newton(DFPN)iterations,and prove that the solutions of the three it-erative methods all converge to the exact solution of the diffusion problem with second-order accuracy in both space and time.It is also proved that the solutions of Picard iteration con-verge linearly to the solution of the discrete scheme,while PN and DFPN iterative solutions converge with a quadratic speed.Numerical tests verify the theoretical results,and highlight the superiority of the TLCD scheme and PN-type iterations.