Research On The Finite Volume Scheme And Conservative Parallel Difference Scheme For Diffusion Equation

In this thesis, we propose some finite volume schemes and conservative paral-lel difference schemes for diffusion equation and give related theoretical analysis to them. First, we construct and analyze some finite volume schemes on dis-torted meshes. When constructing these schemes by using integral interpolation, we need both cell-center and cell-vertex unknowns. In order to reduce the com-putational costs, we present two different methods to eliminate the cell-vertex unknowns. Hence, there are only cell-center unknowns in the resulting finite vol-ume schemes. One is a method of keeping diffusion flux continuity, which gives explicit expression of discrete flux and has local stencil. By using discrete func-tional analysis, we prove that the finite volume scheme is of one order accuracy and confirmed this by numerical experiments. The other method is based on the control volume of vertex. We construct a scheme only containing the cell-vertex unknowns, and we can compute the cell-centered unknowns and the cell-vertex unknowns independently. We give the numerical experiments for linear diffusion problem to show the accuracy of the scheme. Second, we construct some conser-vative parallel difference schemes with intrinsic parallelism for diffusion equation with continuous or discontinuous coefficient, respectively. These schemes are un-conditionally stable, and the rates of their convergence are as same as those of the corresponding fully implicit schemes. At last, we apply the finite volume schemes to application problems. In each chapter, we give some numerical experiments to test the schemes proposed. The numerical results confirm the theoretical predi-cations, and show that these schemes are robust and effective.