Abstract:
In this thesis, an efficient algorithm based on sixth-order compact finite difference and fast discrete sine transform is developed for solving one and two dimensional Poisson equations with Dirichlet boundary conditions. Centered compact finite difference is employed to approximate second order derivatives, and thereafter, each second-order derivative term is replaced by its higher order Taylor series expansion which helps to develop the scheme based on weighted sum and higher order finite differences. This finally results in an implicit compact finite difference scheme which approximates the second order derivatives with six order accuracy. When implementing the method, the developed sixth-order compact difference scheme is applied to discretize the Poisson equation which leads to a large system of linear algebraic equations. The discretization process produces systems consisting of tri- and penta-diagonal matrices. In addition, in the 2D case, we used the Kronecker (Tensor) product to obtain matrices associated with the system. In both cases, an efficient and fast solver is developed based on discrete sine transform which is found to significantly reduce the computational cost for solving large systems and approximate the solution with sixth-order accuracy. We have illustrated numerically that the proposed method converges with sixth-order accuracy coinciding with the theoretical analysis. Moreover, the efficiency of the method is illustrated by solving different test problems. According to our experiments, the proposed sixth-order method provides more accurate solutions than the existing fourth-order compact finite difference method. In addition, by varying the number of gird points from small to large, which is equivalent to varying the mesh-size from large to small, we came to realize that our method is numerically stable.
