Solution of the two dimensional second biharmonic equation with high‐order accuracy

Abstract

If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service information about how to choose which publication to write for and submission guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information. About Emerald www.emeraldinsight.comEmerald is a global publisher linking research and practice to the benefit of society. The company manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well as providing an extensive range of online products and additional customer resources and services.Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. AbstractPurpose -The purpose of this paper is to introduce efficient methods for solving the 2D biharmonic equation with Dirichlet boundary conditions of second kind. This equation describes the deflection of loaded plate with boundary conditions of simply supported plate kind. Also it can be derived from the calculus of variations combined with the variational principle of minimum potential energy. Because of existing fourth derivatives in this equation, introducing high-order accurate methods need to use artificial points. Also solving the resulted linear system of equations suffers from slow convergence when iterative methods are used. This paper aims to introduce efficient methods to overcome these problems. Design/methodology/approach -The paper considers several compact finite difference approximations that are derived on a nine-point compact stencil using the values of the solution and its second derivatives as the unknowns. In these approximations there is no need to define special formulas near the boundaries and boundary conditions can be incorporated with these techniques. Several iterative linear systems solvers such as Krylov subspace and multigrid methods and their combination (with suitable preconditioner) have been developed to compare the efficiency of each method and to design powerful solvers. Findings -The paper finds that the combination of compact finite difference schemes with multigrid method and Krylov iteration methods preconditioned by multigrid have excellent results for the second biharmonic equation, and that Krylov iteration methods preconditioned by multigrid are the most efficient methods. Originality/value -The paper is of value in presenting, via some tables and figures, some numerical experiments which resulted from applying new methods on several test problems, and making comparison with conventional methods.

A Decoupling and Coupling Approach for the Bi-harmonic Equation

“…Therefore the function 𝑣 needs to be approximated from the discrete form of Δ𝑤 = 𝑣 [4][5][6][7]. Moreover, there may also be a need for matrix decomposition and Fast Fourier Transforms with this case [8].It is to be noticed that knowing 𝑣 is of practical importance because it has a physical meaning such as bending moment in plate deflection.…”

A Meshless Method Based on Radial Basis and Spline Interpolation for 2-D and 3-D Inhomogeneous Biharmonic BVPs

“…Also, analysing these types of boundary value problems are important in numerical analysis due to having high derivatives and multiple boundary conditions, so they play an important role in testing numerical procedures. A great deal of research work has been published recently on the development of numerical solutions of biharmonic equations, for instance, the finite difference schemes [1,2], numerical method based on B-spline approximation [3], compact approximation schemes for the Laplace operator of fourth-and sixth-order [4], fourth-order compact difference discretisation scheme with unequal mesh sizes [5], fourth-order finite difference approximation based on arithmetic average discretisation [6], multigrid and preconditioned Krylov iterative methods [7,8], boundary integral equation method [9], a spectral collocation method [10], indirect radial-basis-function collocation method [11], high-order boundary integral equation method [12], Galerkin boundary node method [13], an integral collocation approach based on Chebyshev polynomials [14], a numerical method based on neural-network-based functions [15], a collocation method based on a Cartesian grid [16], fast multiple method [17], and multilevel radial basis functions and domain decomposition methods [18].…”

A compact discretization ofO(h4) for two-dimensional nonlinear triharmonic equations

“…Bauer and Riess [4] have used the block iterative method to solve the equation. Later, Kwon et al [5], Stephenson [6], Mohanty et al [7][8][9][10] and Dehghan and Mohebbi [11,12] developed certain second-and fourth-order finite-difference approximations for the biharmonic problems using a compact cell. Recently, using the coupled approach, Mohanty [13] has discussed nine-point compact finite-difference methods of order two and four for the solution of 2D nonlinear biharmonic problems.…”