||In this thesis, we propose two high-order compact finite difference schemes for solving 1-D unsteady reaction-convection-diffusion problems. For the spatial discretization, the first scheme employs the fourth-order Spotz compact difference formula while the second scheme uses the fourth-order exponential compact difference formula. For discretizing the temporal variable, both schemes utilize the Pade approximation. First, we derive the spatially high-order compact difference schemes for the corresponding steady-state equation with a source term. We then apply the resulting compact difference schemes to the unsteady equation without source terms to obtain the semi-discrete formulation, which is an initial-value problem of a large system of ordinary differential equations. Finally, we apply the Pade approximation to compute the numerical solution of the initial-value problem. Under some assumptions, we prove that both schemes are unconditionally stable. Numerical examples are given to illustrate the effectiveness of the newly proposed compact difference schemes. From the numerical results, we find that for small mesh-Peclet numbers, both schemes achieve fourth-order accuracy in temporal and spatial variables. However, the accuracy of both schemes is deteriorated when the mesh-Peclet number is getting large, and in this case, the second scheme is apparently more accurate than the first scheme.|
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