On Compact Finite Difference Schemes With Applications To Moving Boundary Problems

Compact finite differences are introduced with the purpose of developing compact methods of higher order for the numerical solution of ordinary and elliptic partial differential equations.;The notion of poisedness of a compact finite difference is introduced. It is shown that if the incidence matrix of the underlying interpolation problem contains no odd unsupported sequences then the Polya conditions are necessary and sufficient for poisedness.;A Pade Operator method is used to construct compact formulae valid for uniform three point grids. A second Function-Theoretic method extends compact formulae to variably-spaced three point grids with no deterioration in the order of the truncation error.;A new fourth order compact method (CI4) leading to matrix systems with block tridiagonal structure, is applied to boundary value problems associated with second order ordinary differential equations. Numerical experiments with both linear and nonlinear problems and on uniform and nonuniform grids indicate rates of convergence of four.;An application is considered to the time-dependent one-dimensional nonlinear Burgers\u27 equation in which an initial sinusoidal disturbance develops a very sharp boundary layer. It is found that the CI4 method, with a small number of points placed on a highly stretched grid, is capable of accurately resolving the boundary layer.;A new method (LCM) based on local polynomial collocation and Gauss-type quadrature and leading to matrix systems with block tridiagonal structure, is used to generate high order compact methods for ordinary differential equations. A tenth order method is shown to be considerably more efficient than the CI4 method.;A new fourth order compact method, based on the CI4 method, is developed for the solution, on variable grids, of two-dimensional, time independent elliptic partial differential equations. The method is applied to the ill-posed problem of calculating the interface in receding Hele-Shaw flow. Comparisons with exact solutions indicate that the numerical method behaves as expected for early times.;Finally, in an application to the simulation of contaminant transport within a porous medium under an evolving free surface, new fourth order explicit compact expressions for mixed derivatives are developed

On Weyl-Titchmarsh Theory for Singular Finite Difference Hamiltonian Systems

We develop the basic theory of matrix-valued Weyl-Titchmarsh M-functions and the associated Green's matrices for whole-line and half-line self-adjoint Hamiltonian finite difference systems with separated boundary conditions.Comment: 30 pages, to appear in J. Comput. Appl. Mat

Developing Clean Technology through Approximate Solutions of Mathematical Models

In this paper, the role of mathematical modeling in the development of clean technology has been considered. One method each for obtaining approximate solutions of mathematical models by ordinary differential equations and partial differential equations respectively arising from the modeling of systems and physical phenomena has been considered. The construction of continuous hybrid methods for the numerical approximation of the solutions of initial value problems of ordinary differential equations as well as homotopy analysis method, an approximate analytical method, for the solution of nonlinear partial differential equations are discussed

Fifty Years of Stiffness

The notion of stiffness, which originated in several applications of a different nature, has dominated the activities related to the numerical treatment of differential problems for the last fifty years. Contrary to what usually happens in Mathematics, its definition has been, for a long time, not formally precise (actually, there are too many of them). Again, the needs of applications, especially those arising in the construction of robust and general purpose codes, require nowadays a formally precise definition. In this paper, we review the evolution of such a notion and we also provide a precise definition which encompasses all the previous ones.Comment: 24 pages, 11 figure

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde