The Finite Volume Scheme And Parallel Difference Scheme For Diffusion Equation

In this thesis, we propose some finite volume schemes and parallel difference schemes for diffusion equation and give some theoretical analysis to them. First, we construct and analyze some finite volume schemes on distorted meshes, which include matching mesh and non-matching mesh. When constructing these schemes by using integral interpolation, we need both cell-center and cell-vertex unknowns. In order to reduce the computational costs, we present some different methods to eliminate the cell-vertex unknowns. Hence, there are only cell-center unknowns in the resulting finite volume schemes, and these schemes have explicit expression of discrete flux and have local stencil. Using discrete functional analysis, we prove that the finite volume schemes are of first order accuracy. Then, we construct some parallel difference schemes with intrinsic parallelism for diffusion equation with continuous or discontinuous coefficients, respectively. Moreover, we prove that these schemes are unconditionally stable, and the rates of convergence of them are the same as those of the corresponding fully implicit schemes. At last, we apply some parallel difference schemes to application problems. In each chapter, we give some numerical experiments to test the schemes proposed. The numerical results confirm the theoretical predications, and show that these schemes are robust and effective.