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...

We consider the fractional cable equation. For solution of fractional Cable equation involving Caputo fractional derivative, a new difference scheme is constructed based on Crank Nicholson difference scheme. We prove that the proposed method is unconditionally stable by using spectral stability technique.

A second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for two semidiscretisations in time and a full discretisation using P1 FEM in space. Both the Backward-Euler method and the Crank-Nicolson method are considered, and certain critical details of the implementation are ...

In this paper convergence properties are discussed for some locally one-dimensional (LOD) splitting methods applied to linear parabolic initialboundary value problems. We shall consider unconditional convergence, where both the stepsize in time and the meshwidth in space tend to zero, independently of each other.

Black-Scholes model plays a very significant role in the world of quantitative finance. In this paper, focus are on both nonlinear and linear (BS) equations with numerical approximations. We aim to find an effective approximations for model. Several models from most relevant class European option analyzed study. The problem is approached by transforming into convection-diffusion equation later ...

Numerical solution methods for pricing American options are considered. We propose a second-order accurate Runge-Kutta scheme for the time discretization of the Black-Scholes partial differential equation with an early exercise constraint. We reformulate the algorithm introduced by Brennan and Schwartz into a simple form using a LU decomposition and a modified backward substitution with a proje...

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The schemeuses a non-uniformgrid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show th...

We analyze finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method, at the order of O(h2 + τ2) in the l2-norm and discrete H1-norm with time step τ and mesh size h. Be...

Thermal interaction of fluids and solids, or conjugate heat transfer (CHT), is encountered in many engineering applications. Since time-accurate computations of such coupled problems can be computationally expensive, we consider loosely-coupled and stronglycoupled solution algorithms in which higher order multi-stage Runge-Kutta schemes are employed for time integration. The higher order time i...