Distort The Linear Diffusion Equation Grid Accurate Differential Format

In this thesis we study the linearly exact method for the numerical solution of diffusion equation on distorted grids.As a revelatory approach,the linearly exact method requires that the discretization is exact whenever the control equation possesses a linear solution. In this context,we first prove the numerical stability and convergence of a linearly exact nine-point scheme based on bilinear interpolation,using the discrete functional analysis tools.Then we obtain a linearly exact finite volume scheme based on variational principal for structured grids. With the new scheme we could treat all kinds of boundary conditions in a unified frame.Moreover we reveal certain connection between our new scheme and the support operators scheme.Finally we develop a linearly exact version of the previous variational-principal based finite volume scheme on unstructured triangular grids.Numerical experiments are presented to confirm the robustness and efficiency of the linearly exact schemes.