Global extrapolation procedures for special and general initial value problems

Two- and three- grid global extrapolation procedures are considered fohe special and general initial value problems of arbitrary order r tq. Extrapolation formulas are developed for consistent numerical methods of arbitrary order p . The global extrapolations of a number of existing numerical methods are considered and tested on three problems from the literature

Lo - stable methods for parabolic partial differential equations

In recent years much attention has been devoted in the literature to the development, analysis and implementation of extrapolation methods for the numerical solution of partial differential equations with mixed initial and boundary values specified, see, for example, Lawson and Morris [5], Lawson and Swayne [6] and Gourlay and Morris [3]. The essential theme of these papers was to develop Lo-stable methods for the solution of parabolic partial differential equations in which splitting methods, such as the Crank-Nicolson method, are less than satisfactory when a time discretization is used with time steps which are too large relative to the spatial discretization. In the present paper a family of new Lo-stable methods based on Padé approximants to the exponential function is developed, and. higher accuracy is achieved. The methods are tested on heat equations in one and two space dimensions in which discontinuities exist between the initial and boundary conditions

A family of difference schemes for fourth order parabolic partial differential equations

A family of methods is developed for the numerical solution of fourth order parabolic partial differential equations in one- and two-space variables. The methods are seen to evolve from multiderivative methods for second order ordinary differential equations. The methods are tested on three model problems, with constant coefficients and variable coefficients, which have appeared in the literature

Stability regions for one-step multiderivative methods

Stability regions are plotted for certain members of a family of one-step multiderivative predictor-corrector methods developed by the authors in an earlier paper. The methods discussed are tested on a linear system where the matrix of coefficients has constant complex eigenvalues and on a stiff non-linear system arising in reactor kinetics

Multiderivative methods for periodic initial value problems

A family of two-step multiderivative methods based on Pade approximants to the exponential function is developed. The methods are analysed and periodicity intervals in PECE mode are calculated. Two of the methods are tested on two problems from the literature and one predictor-corrector combination is tested on two further problems

Backward difference replacements of the space derivative in first order hyperbolic equations

Two families of two-time level difference schemes are developed for the numerical solution of first order hyperbolic partial differential equations with one space variable. The space derivative is replaced by (i) a first order, (ii) a second order backward difference approximant and the resulting system of first order ordinary differential equations is solved using A0-stable and L0-stable methods. The methods are tested on a number of problems from the literature involving wave-form solutions, increasing solutions with discontinuities in function values or first derivatives across a characteristic, and exponentially decaying solutions

An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering

In this work we study an interior penalty method for a finite-dimensional large-scale linear complementarity problem (LCP) arising often from the discretization of stochastic optimal problems in financial engineering. In this approach, we approximate the LCP by a nonlinear algebraic equation containing a penalty term linked to the logarithmic barrier function for constrained optimization problems. We show that the penalty equation has a solution and establish a convergence theory for the approximate solutions. A smooth Newton method is proposed for solving the penalty equation and properties of the Jacobian matrix in the Newton method have been investigated. Numerical experimental results using three non-trivial test examples are presented to demonstrate the rates of convergence, efficiency and usefulness of the method for solving practical problems

A Robust Numerical Scheme for Pricing American Options Under Regime Switching Based on Penalty Method

This paper is devoted to develop a robust numerical method to solve a system of complementarity problems (CPs) arising from pricing American options under regime switching. Based on a penalty method, the system of complementarity problems are approximated by a set of coupled nonlinear partial differential equations (PDEs). We then introduce a fitted finite volume (FFVM) method for the spatial discretization along with a fully implicit time stepping scheme for the PDEs, which results in a system of nonlinear algebraic equations. We show that this scheme is consistent, stable and monotone, hence convergent. To solve the system of nonlinear equations effectively, an iterative solution method is established. The convergence of the solution method is shown. Numerical tests are performed to examine the convergence rate and verify the effectiveness and robustness of the new numerical scheme