Research On The Principle Of Positivity-preserving Extremum Principle Format For Convection-diffusion Equation

In this thesis,we consider the following three problems:(1)Monotone fi-nite volume scheme for diffusion equations on general non-conforming meshes;(2)Finite volume scheme preserving extremum principle for convection-diffusion equations on polygonal meshes;(3)Positivity-preserving iterative scheme for im-perfect contact interface problem.First,a nonlinear monotone finite volume scheme on general non-conforming meshes for diffusion equations is introduced,which deals with discontinuous ten-sor coefficients rigorously.Since the expression of normal flux depends on aux-iliary unknowns defined at cell-vertex including hanging nodes,we propose a new method to eliminate vertex-unknown by using primary unknowns at the centers of the cells sharing the vertex.Especially the unknowns defined on hang-ing nodes are eliminated by flux continuous conditions.The resulting scheme is monotone and preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficient problems.Numerical results show that the convergent order of the monotone scheme by different methods of eliminating vertex unknowns will vary remarkably,and our new method can assure that it has almost second order accuracy and more accurate than some existing methods.Second,we present a new nonlinear finite volume scheme preserving posi-tivity for diffusion equations.The main feature of the scheme is the assumption that the values of auxiliary unknowns are nonnegative is avoided.Two nonnega-tive parameters are introduced to define a new nonlinear two-point flux,in which one point is the cell-center and the other is the midpoint of cell-edge.The final flux on the edge is obtained by the continuity of normal flux.Numerical results show that the accuracy of both solution and flux for our new scheme is superior to that of some existing monotone schemes.Then,we propose a nonlinear finite volume scheme for convection-diffusion equation on polygonal meshes and prove that the discrete solution of the scheme satisfies the discrete extremum principle.The approximation of diffusive flux is based on an adaptive approach of choosing stencil in the construction of discrete normal flux,and the approximation of convection flux is based on the second-order upwind method with proper slope limiter.Our scheme is locally conser-vative and has only cell-centered unknowns.Numerical results show that our scheme can preserve discrete extremum principle and has almost second-order accuracy.At last,we introduce a positivity-preserving iterative scheme for imperfect contact interface problem.It is difficult to solve the algebraic system arising from the existing discretization method for imperfect contact interface problem,in which the interface conditions are different from those of perfect contact in-terface problems.To tackle the difficulty,we consider the well-known domain decomposition method in this paper.First we use a kind of iterative domain de-composition method for the model problem,then we apply a positivity-preserving scheme to solve the sub-domain problems with Robin boundary condition on in-ner interface.Moreover,we give the proof of positivity of the weak solution for this kind of interface problem.We also prove that the iterative method is positivity-preserving during the iterative procedure.Numerical results verify that our scheme can preserve positivity of the solutions and has second-order accuracy.