An accurate three spatial grid-point discretization of O( k2+ h4) for the numerical solution of one-space dimensional unsteady quasi-linear biharmonic problem of second kind

• Published in: Applied mathematics and computation Vol. 140; no. 1; pp. 1 - 14
• Main Author: Mohanty, R.K.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 30.07.2003; Elsevier
• Subjects: Exact sciences and technology; Mathematics; Non-linear [formula omitted] equation; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Quasi-linear equation; RMS errors; Sciences and techniques of general use; Singular equation; Two level implicit scheme; Unsteady biharmonic problem; Two level implicit scheme; Quasi-linear equation; RMS errors; Non-linear K dV equation; Singular equation; Unsteady biharmonic problem; Convergence of numerical methods; Implicit method; Initial condition; Quasi linear equation; Unsteady state; Two level method; Boundary condition; Partial differential equation; Discretization method; Korteweg de Vries equation; Mean square error; Non linear equation; Numerical analysis; Biharmonic equation; Finite difference method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/S0096-3003(02)00175-3

In this article, using three spatial-grid points we propose two new two level implicit finite difference approximations of O( k 2+ h 2) and O( k 2+ h 4) in a coupled manner to the one-space dimensional unsteady quasi-linear biharmonic equation A( x, t, u, u xx ) u xxxx + u t = f( x, t, u, u x , u xx , u xxx ), 0< x<1, t>0 subject to the initial and boundary conditions u( x,0)= φ( x), u(0, t)= p 0( t), u xx (0, t)= q 0( t), u(1, t)= p 1( t), u xx (1, t)= q 1( t) are prescribed, where h>0 and k>0 are mesh sizes in x- and t-directions, respectively. The numerical solution of u xx is obtained as a by-product of the method and we do not require to discretize the boundary conditions. The methods are successfully tested on the problems having singularities. Numerical results are provided to demonstrate the convergence of new methods.