Crank Nicolson

Natural convection effects of the numerical solution for unsteady, laminar, free convection flow over an incompressible viscous fluid past a no n-isothermal vertical cone with surface temperature T ' w(x) = T ' ∞ + ax n varying as power function of distance from the apex (x = 0) is presented here. The dimensionless governing equations of the flow that

This project work report provides a full solution of simplified Navier Stokes equations for The Incompressible Couette Problem. The well known analytical solution to the problem of incompressible couette is compared with a numerical solution. In that paper, I will provide a full solution with simple C code instead of MatLab or Fortran codes, which are known. For discrete problem formulation, implicit Crank-Nicolson method was used. Finally, the system of equation (tridiagonal) is solved with both Thomas and simple Gauss Method. Results of both methods are compared.

In this paper a thick hollow cylinder with finite length made of two dimensional functionally graded material (2D-FGM) subjected to transient thermal boundary conditions is considered. The volume fraction distribution of materials, geometry and thermal boundary conditions are assumed to be axisymmetric but not uniform along the axial direction. The finite element method with graded material properties within each element is used to model the structure and the Crank–Nicolson finite difference method is implemented to solve time dependent equations of the heat transfer problem. Two-dimensional heat conduction in the cylinder is considered and variation of temperature with time as well as temperature distribution through the cylinder are investigated. Effects of variation of material distribution in two radial and axial directions on the temperature distribution and time response are studied. The achieved results show that using two-dimensional FGM leads to a more flexible design so that transient temperature, maximum amplitude and uniformity of temperature distributions can be modified to achieve required specifications by selecting a suitable material distribution profile in two directions.

... For example, a finite difference method is used in [2], [15], [16], [24], a finite element method is considered in [28], and a finite volume method in [10]. The Black-Scholes partial differential equation contains variable coefficients for the first-order and second-order spatial ...

We propose a general framework for efficient pricing via a partial differential equation (PDE) approach for exotic cross-currency interest rate (IR) derivatives, with strong emphasis on long-dated foreign exchange (FX) IR hybrids, namely Power Reverse Dual Currency (PRDC) swaps with a FX Target Redemption (FX-TARN) provision. The FX-TARN provision provides a cap on the FX-linked PRDC coupon amounts, and once the accumulated coupon amount reaches this cap, the underlying PRDC swap terminates.Our PDE pricing framework is based on an auxiliary state variable to keep track of the total accumulated PRDC coupon amount. Finite differences on uniform grids and the Alternating Direction Implicit (ADI) method are used for the spatial and time discretizations, respectively, of the model-dependent PDE corresponding to each discretized value of the auxiliary variable. Numerical examples illustrating the convergence properties of the numerical methods are provided.

The paper is devoted to pricing options characterized by discontinuities in the terminal condition. Finite dierence schemes are examined to highlight how discontinuities can generate numerical drawbacks such as spurious oscillations. We propose a nite dierence scheme that is free of spurious oscillations and satises both the positivity requirement and maximum principle, as it is demanded for the nancial and diusiv e solution of the original Black-Scholes partial dieren tial equation. We explore examples of discrete double barrier knock-out call options and the results are in very good agreement with those in the literature. 1. Introduction. In the market of nancial derivatives the most important problem is the so called option valuation problem, i.e. to compute a fair value for the option. In this paper we investigate the use of various nite dierence schemes to non-standard option pricing models, characterized by discontinuities in the terminal/boundary condi- tions. In order to mak...

In this paper a thick hollow cylinder with finite length made of two dimensional functionally graded material (2D-FGM) subjected to transient thermal boundary conditions is considered. The volume fraction distribution of materials, geometry and thermal boundary conditions are assumed to be axisymmetric but not uniform along the axial direction. The finite element method with graded material properties within each element is used to model the structure and the Crank-Nicolson finite difference method is implemented to solve time dependent equations of the heat transfer problem. Two-dimensional heat conduction in the cylinder is considered and variation of temperature with time as well as temperature distribution through the cylinder are investigated. Effects of variation of material distribution in two radial and axial directions on the temperature distribution and time response are studied. The achieved results show that using two-dimensional FGM leads to a more flexible design so that transient temperature, maximum amplitude and uniformity of temperature distributions can be modified to achieve required specifications by selecting a suitable material distribution profile in two directions.

In this work we improve the algorithm of Han and Wu [SIAM J. Numer. Anal. 41 (2003), 2081–2095] for American Options with respect to stability, accuracy and order of computational effort. We derive an exact discrete artificial boundary condition (ABC) for the Crank–Nicolson scheme for solving the Black–Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level. Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions. Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples an...

Abstract. We investigate the stability of a fully-implicit, linearly extrapolated Crank-Nicolson (CNLE) time-stepping scheme for finite element spatial discretization of the Navier-Stokes equations. Although presented in 1976 by Baker and applied and analyzed in various ...

The paper is devoted to pricing options characterized by discontinuities in the terminal condition. Finite dierence schemes are examined to highlight how discontinuities can generate numerical drawbacks such as spurious oscillations. We propose a nite dierence scheme that is free of spurious oscillations and satises both the positivity requirement and maximum principle, as it is demanded for the nancial and