Recent advances in the study of a fourth-order compact scheme for the one-dimensional biharmonic equation

It is well-known that non-periodic boundary conditions reduce considerably the overall accuracy of an approximating scheme. In previous papers the present authors have studied a fourth-order compact scheme for the one-dimensional biharmonic equation. It relies on Hermitian interpolation, using functional values and Hermitian derivatives on a three-point stencil. However, the fourth-order accuracy is reduced to a mere first-order near the boundary. In turn this leads to an almost third-order accuracy of the approximate solution. By a careful inspection of the matrix elements of the discrete operator, it is shown that the boundary does not affect the approximation, and a full (optimal) fourth-order convergence is attained. A number of numerical examples corroborate this effect.