Compact Difference Scheme with High Accuracy for One-Dimensional Unsteady Quasi-Linear Biharmonic Problem of Second Kind: Application to Physical Problems (original) (raw)
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A highly accurate numerical solution of a biharmonic equation
Numerical Methods for Partial Differential Equations, 1997
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/∂n 2 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(h 6) on a square mesh (hx = hy = h) and of the fourth-order O(h 4 x , h 2 x h 2 y , h 4 y) 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.
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In this paper, we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three-dimensional non-linear biharmonic partial differential equations on a 19-point compact stencil using coupled approach. The numerical solutions of unknown variable uðx; y; zÞ and its Laplacian r 2 u are obtained at each internal grid point. The resulting stencil algorithm is presented which can be used to solve many physical problems. The proposed method allows us to use the Dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. The new method is tested on three problems and the results are compared with the corresponding second order approximation, which we also discuss using coupled approach.
Some Difference Schemes for the Biharmonic Equation
SIAM Journal on Numerical Analysis, 1975
The Dirichlet problem for biharmonic equation in a rectangular region is considered. The method of splitting is used and two classes of finite difference approximations are defined. Two semi-iterative procedures are considered for obtaining the solution of the resulting coupled system of algebraic equations. It is shown that the rate of convergence of the iterative procedures depends upon the choice of the difference approximation. Estimates for optimum iteration parameters are given and several comparisons are made. An attempt is made to unify the ideas on the splitting technique for solving the first biharmonic boundary value problem.
Journal of Scientific Computing, 2012
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.
High accuracy solution of three-dimensional biharmonic equations
Numerical Algorithms, 2002
In this paper, we consider several nite di erence approximations for the threedimensional biharmonic equation. A symbolic algebra package is utilized to derive a family of nite di erence approximations for the biharmonic equation on a 27 point compact stencil. The unknown solution and its rst derivatives are carried as unknowns at selected grid points. This formulation allows us to incorporate the Dirichlet boundary conditions automatically and there is no need to de ne special formulas near the boundaries, as is the case with the standard discretizations of biharmonic equations.
Explicit block iterative method for the solution of the biharmonic equation
Numerical Methods for Partial Differential Equations, 1993
We develop an iterative algorithm for the solution of a finite difference approximation of the biharmonic equation over a rectangular region by using the explicit block iterative method, i.e., box over relaxation scheme. The I-and 9-point explicit blocks were considered and performance results for the two algorithms are presented.O 1993 John Wiley & Sons, Inc.
Generalized finite difference method for two-dimensional shallow water equations
Engineering Analysis with Boundary Elements, 2017
A novel meshless numerical scheme, based on the generalized finite difference method (GFDM), is proposed to accurately analyze the two-dimensional shallow water equations (SWEs). The SWEs are a hyperbolic system of first-order nonlinear partial differential equations and can be used to describe various problems in hydraulic and ocean engineering, so it is of great importance to develop an efficient and accurate numerical model to analyze the SWEs. According to split-coefficient matrix methods, the SWEs can be transformed to a characteristic form, which can easily present information of characteristic in the correct directions. The GFDM and the second-order Runge-Kutta method are adopted for spatial and temporal discretization of the characteristic form of the SWEs, respectively. The GFDM is one of the newly-developed domain-type meshless methods, so the time-consuming tasks of mesh generation and numerical quadrature can be truly avoided. To use the moving-least squares method of the GFDM, the spatial derivatives at every node can be expressed as linear combinations of nearby function values with different weighting coefficients. In order to properly cooperate with the split-coefficient matrix methods and the characteristic of the SWEs, a new way to determine the shape of star in the GFDM is proposed in this paper to capture the wave transmission. Numerical results and comparisons from several examples are provided to verify the merits of the proposed meshless scheme. Besides, the numerical results are compared with other solutions to validate the accuracy and the consistency of the proposed meshless numerical scheme.
In this study, a high-order compact scheme for 2D Laplace and Poisson equations under a non-uniform grid setting is developed. Based on the optimal difference method, a nine-point compact difference scheme is generated. Difference coefficients at each grid point and source term are derived. This is accomplished through the consideration of compatibility between the partial differential equation and its difference discretization. Theoretically, the proposed scheme has third-to fourth-order accuracy; its fourth-order accuracy is achieved under uniform grid settings. Two examples are provided to examine performance of the proposed scheme. Compared with the traditional five-point difference scheme, the proposed scheme can produce more accurate results with faster convergence. Another reference scheme with the same nine-point grid stencil is derived based on the five-point scheme. The two nine-point schemes have the same coefficients for each grid points; however, their coefficients for the source term are different. The overall accuracy level of the solution resulting from the proposed scheme is higher than that of the nine-point reference scheme. It is also indicated that the smoothness of grids has significant effects on accuracy and convergence of the solutions; efforts in optimizing the grid configuration and allocation can improve solution accuracy and efficiency. Consequently, with the proposed method, solution under the non-uniform grid setting with appropriate grid allocation would be more accurate than that under the uniform-grid manipulation, with the same number of grid points.
Numerische Mathematik, 1971
The numerical solution of the 2-dimensional biharmonic equation over the unit square by using Extrapolated Alternating Direction Implicit (E.A.D.I.) methods is studied. To approximate the biharmonic equation both a t 3-point and a 25-point difference replacements are considered. In each case E.A.D.I. schemes are used together with the acceleration parameter r fixed during the iterations or varying according to the Douglas set of parameters. Finally optimum E.A.D.I. schemes are given for every value of the number N of mesh subdivisions in each coordinate direction.