A Highly Accurate Numerical Solution of a Biharmonic Equation

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24 Scopus citations

Abstract

The coefficients for a nine-point high-order accuracy discretization scheme for a biharmonic equation ∇4 u = f(x, y) (∇2 is the two-dimensional Laplacian operator) are derived. The biharmonic problem is defined on a rectangular domain with two types of boundary conditions: (1) u and ∂2 u/∂n2 or (2) u and ∂u/∂n (where ∂/∂n is the normal to the boundary derivative) are specified at the boundary. For both considered cases, the truncation error for the suggested scheme is of the sixth-order O(h6) on a square mesh (hx = hy = h) and of the fourth-order O(h4x, h2xh2y, h4y) on an unequally spaced mesh. The biharmonic equation describes the deflection of loaded plates. The advantage of the suggested scheme is demonstrated for solving problems of the deflection of rectangular plates for cases of different boundary conditions: (1) a simply supported plate and (2) a plate with built-in edges. In order to demonstrate the high-order accuracy of the method, the numerical results are compared with exact solutions.

Original languageEnglish
Pages (from-to)375-391
Number of pages17
JournalNumerical Methods for Partial Differential Equations
Volume13
Issue number4
DOIs
StatePublished - 1 Jan 1997

Keywords

  • Biharmonic equation
  • Finite difference
  • High-order accuracy

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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