A High-Order Compact Boundary Value Method for Solving One-Dimensional Heat Equations

We combine fourth-order boundary value methods (BVMs) for discretizing the temporal variable with fourth-order compact difference scheme for discretizing the spatial variable to solve one-dimensional heat equations. This class of new compact difference schemes achieve fourth-order accuracy in both temporal and spatial variables and are unconditionally stable due to the favorable stability property of BVMs. Numerical results are presented to demonstrate the accuracy and efficiency of the new compact difference scheme, compared to the standard second-order Crank-Nicolson scheme.