We interpret the Cayley transform of linear (finiteor infinite-dimensional) state space systems as a numerical integration scheme of Crank–Nicolson type. The scheme is known as Tustin’s method in the engineering literature, and it has the following important Hamiltonian integrator property: if Tustin’s method is applied to a conservative (continuous time) linear system, then the resulting (disc...

Several nonlinear Black-Scholes models have been proposed to take transaction cost, large investor performance and illiquid markets into account. One of the most comprehensive models introduced by Barles and Soner in [4] considers transaction cost in the hedging strategy and risk from an illiquid market. In this paper, we compare several finite difference methods for the solution of this model ...

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 develop systematically a numerical approximation strategy to discretize a hydrodynamic phase field model for a binary fluid mixture of two immiscible viscous fluids, derived using the generalized Onsager principle that warrants not only the variational structure but also the energy dissipation property. We first discretize the governing equations in space to arrive at a semi-discretized, tim...

Transparent boundary conditions (TBCs) for general Schrödinger– type equations on a bounded domain can be derived explicitly under the assumption that the given potential V is constant on the exterior of that domain. In 1D these boundary conditions are non–local in time (of memory type). Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nicolson finite d...

The standard ‘parabolic’ approximation to the Helmholtz equation is used in order to model long-range propagation of sound in the sea in the presence of cylindrical symmetry in a domain with a rigid bottom of variable topography. The rigid bottom is modeled by a homogeneous Neumann condition and a paraxial approximation thereof proposed by Abrahamsson and Kreiss. The resulting initial-boundary-...

In this paper, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H . We consider two cases. If H > 1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which 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...

We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order erro...

In this paper, we develop the Crank-Nicolson finite difference method (C-N-FDM) to solve the linear time-fractional diffusion equation, formulated with Caputo’s fractional derivative. Special attention is given to study the stability of the proposed method which is introduced by means of a recently proposed procedure akin to the standard Von-Neumann stable analysis. Some numerical examples are ...