Highly Accurate Implicit Schemes Using Hexagonal Grids for the Approximation of the Derivatives of the Solution of Two Dimensional Heat Equation

In this thesis, the first type (Dirichlet) boundary value problem for the heat equation on a rectangle is considered. The research has two main successes. Firstly, we give a two-stage implicit method of second order accuracy for the approximation of the first order derivatives of the solution with respect to the spatial variables. To approximate the solution at the first stage, the unconditionally stable two layer implicit method on hexagonal grids given by Buranay and Arshad in 2020 is used which converges with second order in space and time variable on the grids. At the second stage, for the approximation of first derivatives with respect to the spatial variables we propose special difference boundary value problems on hexagonal grids of which the boundary conditions are defined by using the obtained solution from the first stage. Further, uniform convergence of the solution of the constructed special difference boundary value problems to the corresponding exact derivatives on hexagonal grids with second order is shown. Secondly, we give fourth order accurate implicit methods for the computation of the first order spatial derivatives and second order mixed derivatives involving the time derivative of the solution. These methods are constructed based on two stages: At the first stage of the methods, the solution is approximated by using the implicit scheme given by Buranay and Arshad in 2020 that gives fourth order of convergence in space and first order in time variables to the exact solution on the constructed hexagonal grids. For the approximation of the derivative of the solution to the heat equation with respect to the time variable an analogous scheme is devised. Subsequently, to approximate the first order spatial derivatives and the second order mixed derivatives of the solution difference boundary value problems on hexagonal grids are constructed at the second stages. Further, uniform convergence of these implicit schemes to the corresponding exact derivatives are shown. Eventually, the developed second order and fourth order accurate two-stage implicit methods are used to solve some test problems and the numerical results illustrating the applicability and the accuracy of the methods are presented through tables and figures. Keywords: Finite difference method; Hexagonal grid; Stability analysis; Two dimensional heat equation; Approximation of derivatives.