Many temporal discretization methods for linear evolution equations converge uniformly on compact time intervals at the rate 1 nα only for sufficiently smooth initial data. It is shown that these methods can be regularized such that the new schemes converge ‘in the average’ at the rate 1 nα for all initial data. Examples given include the Crank-Nicholson scheme and the alternating direction imp...

In this paper we consider a passive scalar transported in two-dimensional flow. The governing equation is that of the convection–diffusion–reaction equation. For purposes of computational efficiency, we apply an alternating-direction implicit scheme akin to that proposed by Polezhaev. Use of this implicit operator-splitting scheme allows the application of a tridiagonal Thomas solver to obtain ...

Maximum drawdown is a risk measure that plays an important role in portfolio management. In this paper, we address the question of computing the expected value of the maximum drawdown using a partial differential equation (PDE) approach. First, we derive a two-dimensional convection-diffusion pricing equation for the maximum drawdown in the Black-Scholes framework. Due to the properties of the ...

Based on the combined compact difference scheme, an alternating direction implicit method is proposed for solving two-dimensional cubic nonlinear Schrödinger equations. The proposed method is sixth-order accurate in space and second-order accurate in time. The linear Fourier analysis method is exploited to study the stability of the proposed method. The efficiency and accuracy of the proposed m...

Alternating Directions Implicit (ADI) integration is an operator splitting approach to solve parabolic and elliptic partial differential equations in multiple dimensions based on solving sequentially a set of related one-dimensional equations. Classical ADI methods have order at most two, due the errors. Moreover, when time discretization stiff problems Runge-Kutta schemes, additional reduction...

A new methodology is introduced for the numerical solution of Partial Differential Equations in general spatial domains. The methodology is based on the use of the well-known Alternating Direction Implicit (ADI) approach of Peaceman and Rachford in conjunction with one-dimensional and high-order accurate Fourier representations of non-periodic data, obtained by way of a certain “continuation me...

We propose an algorithm for the numerical solution of the Lur’e equations in the bounded real and positive real lemma for stable systems. The recently developed ADI iteration for algebraic Riccati equations is generalized to Lur’e equations. The algorithm provides approximate solutions in low-rank factored form. We prove that the sequence of approximate solutions is monotonically increasing wit...

The fundamental idea behind all multigrid methods is to compute using diierent scales in order to obliterate error components of diierent scales. Error components can be thought of as waves. A long wave, which has low frequency, on one grid is a short wave, or high frequency, on a coarse enough related grid. A number of iterative methods (e.g., relaxation and alternating direction implicit meth...

Inspired from the Identity-based cryptosystem proposed by Adi Shamir, and Boneh and Franklin, this paper designed a new Identity-based digital signature (ECM-IDS) scheme using extended chaotic maps. The ECM-IDS scheme is secure based on the difficulties of integer factorization problem.