Compact Difference Schemes For Heat Equation With Neumann Boundary Conditions

W. Y. Liao and J. P. Zhu (J. Comp. Math.,(2005)) derives a high order difference scheme for the following one-dimensional heat equation.but there is no theoretical analysis. In this paper, we prove that the scheme is unconditionally stable, and is convergent with the convergence order of O(Ï„² + h^3.5). Moreover, we improve the scheme and derive a more accurate difference scheme by reducing the error produced by discretization at the boundaries. We also prove that it is absolutely stable and convergent with the convergence order of O(Ï„² + h⁴). Numerical examples testify the theoretical results.In the second part, we study the one-dimensional heat equation with Neumann boundary conditions by the method of reduction of order and Keller Box scheme. By introducing a new variable v = w_x, we eliminate the error produced by the discretization for the derivative on boundaries, derive a fourth-order difference scheme, and analyze existence, uniqueness, convergence and stability of difference solution. A numerical example demonstrates the theoretical results.