A Monotone Correction For The Finite Volume Scheme Of Diffusion Equation On Distorted Meshes

In this thesis,we propose a monotone correction for the finite volume scheme of diffusion equation on distorted meshes.First,we construct a new monotone finite volume scheme for quadrilateral grids based on the original nine-point scheme.The basic idea of the scheme is to divide the terms with vertex unknowns into two terms,then we construct a nonlinear two-point flux based on the positive and negative parts.The main feature of the scheme is the assumption that the values of auxiliary unknowns are nonnegative is avoided.At the same time,we can obtain the expression of the normal flux which has only cell-centered unknowns.Then,we make a modification to the multi-point flux approximation schemes,in this way we can obtain a new nonlinear finite volume scheme preserving positivity for diffusion equations.We take the terms with two adjacent cell-centered unknowns in the flux expression of the multi-point flux approximation scheme firstly,then regard the rest of the terms as a whole.We divide them into the positive and negative parts,and transform them into terms with only two cell-centered unknowns.Finally,we obtain a nonlinear two-point flux formula.Thereby a new nonlinear finite volume scheme preserving positivity for diffusion equations can be obtained.Numerical results demonstrate that our scheme preserves positivity of the solution,which gives almost second-order convergent rate for the solution and the first-order convergent rate for the flux.Comparing with the existing monotone finite volume schemes,our new scheme can not only maintain the high accuracy but also improve the computational efficiency.Hence,we can conclude that our new scheme is superior to some existing monotone finite volume schemes.