We study the Crank–Nicolson scheme for stochastic differential equations (SDEs) driven by a multidimensional fractional Brownian motion with Hurst parameter H>1/2. It is well known that ordinary proper conditions on regularity of coefficients, achieves convergence rate n−2, regardless dimension. In this paper we show that, due to interactions between driving processes, corresponding m-dimension...

*Correspondence: [email protected] 2School of Science, Jiangnan University, Lihu Road, Wuxi, 214122, China Full list of author information is available at the end of the article Abstract In the study of pattern formation in bi-stable systems, the extended Fisher-Kolmogorov (EFK) equation plays an important role. In this paper, a Fourier pseudo-spectral method for solving the EFK equation in ...

An unconditionally stable finite difference scheme for systems whose dynamics are described by a fourth-order partial differential equation is developed with the use of a regular hexagonal grid. The scheme is motivated by the well-known Crank-Nicolson discretization that was originally developed for second-order systems and it is used to develop a discrete in time and space model of a deformabl...

Keywords: Benjamin–Bona–Mahony equation Method of compactness Moving boundary Crank–Nicolson method a b s t r a c t In this work we present the existence, the uniqueness and numerical solutions for a mathematical model associated with equations of Benjamin–Bona–Mahony type in a domain with moving boundary. We apply the Galerkin method, multiplier techniques, energy estimates and compactness res...

Tasks. Implement the implicit finite difference scheme (i.e., backward Euler scheme) and the Crank-Nicolson scheme and compute the option prices in this way. Moreover, study the empirical convergence rates. You need to choose the following numerical parameters: • The truncation values for the infinite domain xmin < 0 < xmax. To simplify the analysis, you should use the same values for all the r...

In this paper we consider a backward parabolic partial differential equation, called the Black-Scholes equation, governing American and European option pricing. We present a numerical method combining the Crank-Nicolson method in the time discretization with a hybrid finite difference scheme on a piecewise uniform mesh in the spatial discretization. The difference scheme is stable for the arbit...

We consider a second-order conservative nonlinear numerical scheme for the Ncomponent Cahn–Hilliard system modeling the phase separation of a N-component mixture. The scheme is based on a Crank–Nicolson finite-difference method and is solved by an efficient and accurate nonlinear multigrid method. We numerically demonstrate the second-order accuracy of the numerical scheme. We observe that our ...

We study accuracy of alternating direction implicit (ADI) methods for parabolic equations. The original ADI method applied to parabolic equations is a perturbation of the Crank-Nicolson di erence equation and has second-order accuracy both in space and time. The perturbation error is on the same order as the discretization error, in terms of mathematical description. However, we often observe i...

We present two unconditionally stable finite-difference time-domain (FDTD) methods for modeling the Sagnac effect in rotating optical microsensors. The methods are based on the implicit Crank-Nicolson scheme, adapted to hold in the rotating system reference frame-the rotating Crank-Nicolson (RCN) methods. The first method (RCN-2) is second order accurate in space whereas the second method (RCN-...

and Applied Analysis 3 As in the classical Crank-Nicholson difference scheme, we will obtain a discrete approximation to the fractional derivative ∂U t, x /∂t at tn 1/2 , xi . Let H t, x 1 Γ 1 − α ∫ t 0 u s, x − u 0, x t − s α ds. 2.1