Recent advances in the study of a fourth-order compact scheme for the one-dimensional biharmonic equation. (English) Zbl 1254.65117
Summary: 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.
MSC:
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
Keywords:
discrete biharmonic operator; nonhomogeneous boundary conditions; fourth-order convergence; Hermite interpolation; compact schemesReferences:
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