Nonstandard finite difference equations for ODEs and 1-D PDEs based on piecewise linearization (Q1126587)

The authors construct nonstandard finite difference schemes for nonlinear ordinary and one-dimensional partial differential equations. The method of construction consists in employing piecewise linearization, which reduces the nonlinear equation to a piecewise one. The latter can be integrated analytically subject to appropriate continuity conditions for both its solution and the first-order derivative of the solution in order to get three-point nonlinear difference schemes, the solution of which provides the nodal values of the approximate solution. The solution of the nonlinear difference schemes requires the use of iterative techniques.