We present a novel numerical method and algorithm for the solution of the 3D axially symmetric timedependent Schrödinger equation in cylindrical coordinates, involving singular Coulomb potential terms besides a smooth time-dependent potential. We use fourth order finite difference real space discretization, with special formulae for the arising Neumann and Robin boundary conditions along the sy...

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...

We construct and analyze a least-squares procedure for approximately solving the initial-value problem for the linearized equations of coupled sound and heat flow, in a bounded domain Í2 in Ä , with homogeneous Dirichlet boundary conditions. The method is based on Crank-Nicolson time differencing. To approximately solve the resulting system of boundary value problems at each time step, a least-...

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 ...

Classical alternating direction (AD) and fractional step (FS) methods for parabolic equations, based on some standard implicit time stepping procedure such as Crank-Nicolson, can have errors associated with the AD or FS perturbations that are much larger than the errors associated with the underlying time stepping procedure. We show that minor modi cations in the AD and FS procedures can virtua...

This paper proposes the use of the Spectral method to simulate diffusive moisture transfer through porous materials, which can be strongly nonlinear and can significantly affect sensible and latent heat transfer. An alternative way for computing solutions by considering a separated representation is presented, which can be applied to both linear and nonlinear diffusive problems, considering hig...

Based on eight saul’yev asymmetry schemes and the concept of domain decomposition, a class of finite difference method (AGE) with intrinsic parallelism for 1D diffusion equations is constructed. Stability analysis for the method is done. We also pay attention to the implementation of the parallel algorithms for 2D convectiondiffusion equations. Based on another group of saul’yev asymmetry schem...