In this paper, we construct a new numerical algorithm for the partial differential equation of up-and-out put barrier options under CEV model. method, use Crank-Nicolson scheme to discrete temporal variables and cubic B-spline collocation method spatial variables. The is stable has second-order convergence both time space analysis proposed discussed in detail. Finally, examples verify stability...

This paper is concerned with moving mesh finite difference solution of partial differential equations. It is known that mesh movement introduces an extra convection term and its numerical treatment has a significant impact on the stability of numerical schemes. Moreover, many implicit second and higher order schemes, such as the Crank-Nicolson scheme, will loss their unconditional stability. A ...

A finite difference method which is second-order accurate in time and in space is proposed for two-dimensional fractional percolation equations. Using the Fourier transform, a general approximation for the mixed fractional derivatives is analyzed. An approach based on the classical Crank-Nicolson scheme combined with the Richardson extrapolation is used to obtain temporally and spatially second...

A new method is formulated and analyzed for the approximate solution of a twodimensional time-fractional diffusion-wave equation. In this method, orthogonal spline collocation is used for the spatial discretization and, for the time-stepping, a novel alternating direction implicit (ADI) method based on the Crank-Nicolson method combined with the L1-approximation of the time Caputo derivative of...

We present high-order compact schemes for a linear second-order parabolic partial differential equation (PDE) with mixed second-order derivative terms in two spatial dimensions. The schemes are applied to option pricing PDE for a family of stochastic volatility models. We use a nonuniform grid with more grid-points around the strike price. The schemes are fourth-order accurate in space and seco...

and Applied Analysis 3 3. Numerical Solution Environmental phenomena, such as groundwater flow described by variational order derivative, are highly complex phenomena, which do not lend themselves readily to analysis of analytical models. The discussion presented in this section will therefore be devoted to the derivation of numerical solution to the modified Hantush equation (6). Solving diffi...

Abstract. We derive optimal order a posteriori error estimates for time discretizations by both the Crank–Nicolson and the Crank–Nicolson–Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second order Crank–Nicolson reconstructions of the piecewise linear approximate solutions. The...

We consider a model initial and boundary value problem for the wide-angle ‘parabolic’ equation Lur = icu of underwater acoustics, where L is a second-order differential operator in the depth variable z with depthand range-dependent coefficients. We discretize the problem by the Crank–Nicolson finite difference scheme and also by the forward Euler method using nonuniform partitions both in depth...

We investigate the stability of a fully-implicit, linearly extrapolated Crank-Nicolson (CNLE) time-stepping scheme for finite element spatial discretization of the Navier-Stokes equations. Although presented in 1976 by Baker and applied and analyzed in various contexts since then, all known convergence estimates of CNLE require a time-step restriction. We propose a new linear extrapolation of t...