Enhancing finite difference approximations for double barrier options: mesh optimization and repeated Richardson extrapolation

We show that the performances of the finite difference method for double barrier option pricing can be strongly enhanced by applying both a repeated Richardson extrapolation technique and a mesh optimization procedure. In particular, first we construct a space mesh that is uniform and aligned with the discontinuity points of the solution being sought. This is accomplished by means of a suitable transformation of coordinates, which involves some parameters that are implicitly defined and whose existence and uniqueness is theoretically established. Then, a finite difference scheme employing repeated Richardson extrapolation in both space and time is developed. The overall approach exhibits high efficacy: barrier option prices can be computed with accuracy close to the machine precision in less than one second. The numerical simulations also reveal that the improvement over existing methods is due to the combination of the mesh optimization and the repeated Richardson extrapolation.

Approximate Solution of Volterra Integro-Fractional Differential Equations Using Quadratic Spline Function

In this paper, we suggest two new methods for approximating the solution to the Volterra integro-fractional differential equation (VIFDEs), based on the normal quadratic spline function and the second method used the Richardson Extrapolation technique the usage of discrete collocation points. The fractional derivatives are regarded in the Caputo perception. A new theorem for the Richardson Extrapolation points for using the finite difference approximation of Caputo derivative is introduced with their proof. New techniques using the first derivative at the initial point such that obtained by follow two cases the first using trapezoidal rule and the second using the first step of linear spline function using the Richardson Extrapolation method. Specifically, the program is given in examples analysis in Matlab (R2018b). Numerical examples are available to illuminate the productivity and trustworthiness of the methods, as well as, follow the Clenshaw Curtis rule for calculating the required integrals for those equations.

Stagnation Point Flow with Time-Dependent Bionanofluid Past a Sheet: Richardson Extrapolation Technique

The study of laminar flow of heat and mass transfer over a moving surface in bionanofluid is of considerable interest because of its importance for industrial and technological processes such as fabrication of bio-nano materials and thermally enhanced media for bio-inspired fuel cells. Hence, the present work deals with the unsteady bionanofluid flow, heat and mass transfer past an impermeable stretching/shrinking sheet. The appropriate similarity solutions transform the boundary layer equations with three independent variables to a system of ordinary differential equations with one independent variable. The finite difference coupled with the Richardson extrapolation technique in the Maple software solves the reduced system, numerically. The rate of heat transfer is found to be higher when the flow is decelerated past a stretching sheet. It is understood that the state of shrinking sheet limits the rate of heat transfer and the density of the motile microorganisms in the stagnation region.

A second order numerical method for a class of parameterized singular perturbation problems on adaptive grid

A nonlinear singularly perturbed boundary value problem depending on a parameter is considered. First, we solve the problem using the backward Euler finite difference scheme on an adaptive grid. The adaptive grid is a special nonuniform mesh generated through equidistribution principle by a positive monitor function depending on the solution. The behavior of the solution, the stability and the error estimates are discussed. Then, the Richardson extrapolation technique is applied to improve the accuracy of the computed solution associated to the backward Euler scheme. The proofs of the uniform convergence for the backward Euler scheme and the Richardson extrapolation are carried out. Numerical experiments validate the theoretical estimates and indicates that the estimates are sharp.

Unsteady MHD mixed convection flow past an oscillating plate with heat source/sink

This paper to study unsteady MHD mixed convection flow past an infinite vertical oscillating plate through porous medium, taking account of the presence of free/forced convection and mass transfer. Using similarity transformation, the coupled non – linear governing equations are solved numerically by applying the combination of the base scheme submethods – midpoint, and a method enhancement scheme Richardson extrapolation technique together with Fehlberg fourth-fifth order Runge-Kutta shooting iteration method with degree four interpolant. The results are obtained for velocity, temperature, concentration. The effects of various material parameters are discussed on flow variables and presented by graphs.DOI: http://dx.doi.org/10.3329/jname.v11i2.6477

Richardson Cascadic Multigrid Method for 2D Poisson Equation Based on a Fourth Order Compact Scheme

Based on a fourth order compact difference scheme, a Richardson cascadic multigrid (RCMG) method for 2D Poisson equation is proposed, in which the an initial value on the each grid level is given by the Richardson extrapolation technique (Wang and Zhang (2009)) and a cubic interpolation operator. The numerical experiments show that the new method is of higher accuracy and less computation time.