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...

The time dependent Ginzburg-Landau (TDGL) equation is a typical model in phase field theory for many applications like two phase flow simulations and phase transitions. In this paper, we develop effective algorithms so that the solution of the TDGL model can be accurately approximated. Specifically, we adopt finite element methods for the spatial discretization and study different algorithms fo...

We consider an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann–Liouville type and order α ∈ (1, 2). We study a spatial semidiscrete scheme using the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and the Crank...

This paper addresses the construction of absorbing boundary conditions for the one-dimensional Schrödinger equation with a general variable repulsive potential or with a cubic nonlinearity. Semi-discrete time schemes, based on Crank-Nicolson approximations, are built for the associated initial boundary value problems. Finally, some numerical simulations give a comparison of the various absorbin...

The Trapezoidal Rule with second order Backward Difference Formula (TR-BDF2) time stepping method was applied to the Black-Scholes PDE for option pricing. It is proved that TR-BDF2 time stepping method is unconditionally stable, and compared to the usual Crank-Nicolson time stepping method, the TR-BDF2 shows fewer oscillations when computing the derivatives of the solution, which are important ...

In this paper, a three-stage fourth-order numerical scheme is proposed. The first and second stages of the proposed are explicit, whereas third stage implicit. A compact considered to discretize space-involved terms. stability in space time checked using von Neumann criterion for scalar case. region obtained by more than one given explicit Runge–Kutta methods. convergence conditions found syste...

In this paper, we use the finite difference methods to explore step-down Equity Linked Securities (ELS) options under fractional Black-Scholes model. We establish Crank-Nicolson scheme one asset and study impact of Hurst exponent (H) on return repayment fixed stock price. also price different H. Through numerical experiments, it is found that related H, result will increase with case two assets...

Recently, an implicit, nonlinearly consistent, energyand charge-conserving particle-incell method has been proposed for multi-scale, full-f kinetic electrostatic simulations [1]. The method employs a Jacobian-free Newton–Krylov (JFNK) solver, capable of using very large timesteps of field evolution without loss of numerical stability or accuracy. A fundamental feature of the method is the nonli...

We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schr\"odinger-Poisson system. More specifically, we use Crank-Nicolson as stepping mechanism, whilst nonlinearity is handled by means relaxation approach spirit \cite{Besse, KK} nonlinear Schr\"odinger equation. For spatial discretisation standard conforming finite element sch...