Research On Conservative Positivity Persevering Repair Algorithm And Positivity-preserving Scheme For Diffusion Equation

In this thesis, we consider the following two problems:(1) The monotone finite volume scheme for the elliptic interface problems. (2) The conservative enforcing positivity-preserving algorithms for structured and unstructured meshes, respectively.First, we construct a monotone finite volume scheme for the elliptic interface problems. The body-fitted structured quadrilateral mesh is used to divide the computational domain. Two definitions of the discrete flux for the interface edges are proposed and both of them are proved to be monotone. The numerical results show that the method 2 is able to capture the discontinuities of unknown on the interface, it obtains second order convergent rate for L2 and L∞ norm errors of unknown and first order convergent rate for L2 norm error of flux on both the rectangular and random meshes.Second, in view of the convectional diffusion schemes conflict with the discrete extremum principle,(such as the Kershaw scheme, the nine-points scheme,etc),there often appear overshoots and undershoots in the numerical solutions. Taking the advantages of the structured mesh, we propose a conservative enforcing negative to zero algorithm(CENZ) based on the dimension split idea. The algorithm is an improvement of the traditional enforcing negative values to zero (ENZ). It not only guarantees the non-negative of numerical solution but also keeps the conser-vation of total energy and the local conservation of flux. Numerical experiments show that the accuracy and efficiency of this method are better than those of some existing repair methods. It can be applied to repair any numerical results from the finite volume schemes which conflict with the discrete maximum principle.At last, for a class of diffusion schemes without preserving positivity on un-structured mesh, we propose a general conservative enforcing negative values to zero algorithm(GCENZ). The algorithm guarantees the repaired numerical solu-tions to be non-negative and keeps the conservation of total energy and the local conservation of flux. Numerical experiments demonstrate that this algorithm is efficient for repairing the negative values and the conservative errors are much smaller than that of the ENZ method.