Search results for "finite difference"
showing 10 items of 122 documents
Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation
2004
A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.
Numerical approximation of the viscous quantum hydrodynamic model for semiconductors
2006
The viscous quantum hydrodynamic equations for semiconductors with constant temperature are numerically studied. The model consists of the one-dimensional Euler equations for the electron density and current density, including a quantum correction and viscous terms, coupled to the Poisson equation for the electrostatic potential. The equations can be derived formally from a Wigner-Fokker-Planck model by a moment method. Two different numerical techniques are used: a hyperbolic relaxation scheme and a central finite-difference method. By simulating a ballistic diode and a resonant tunneling diode, it is shown that numerical or physical viscosity changes significantly the behavior of the solu…
Efficient numerical methods for pricing American options under stochastic volatility
2007
Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two-dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M-matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved u…
Smoothed Particle ElectroMagnetics: A mesh-free solver for transients
2006
AbstractIn this paper an advanced mesh-free particle method for electromagnetic transient analysis, is presented. The aim is to obtain efficient simulations by avoiding the use of a mesh such as in the most popular grid-based numerical methods. The basic idea is to obtain numerical solutions for partial differential equations describing the electromagnetic problem by using a set of particles arbitrarily placed in the problem domain. The mesh-free smoothed particle hydrodynamics method has been adopted to obtain numerical solution of time domain Maxwell's curl equations. An explicit finite difference scheme has been employed for time integration. Details about the numerical treatment of elec…
Numerical experiments with single mode gyrotron equations
2012
Gyrotrons are microwave sources whose operation is based on the stimulated cyclotron radiation of electrons oscillating in a static magnetic field. This process is described by the system of two complex differential equations: nonlinear first order ordinary differential equation with parameter (averaged equation of electron motion) and second order partial differential equation for high frequency field (RF field) in resonator (Schrödinger type equation for the wave amplitude). The stationary problem of the single mode gyrotron equation in short time interval with real initial conditions was numerically examined in our earlier work. In this paper we consider the stationary and nonstationary …
EFFECTIVE FINITE-DIFFERENCE METHODS FOR THE SOLUTIONS OF FILTRATION PROBLEMS IN MULTILAYER DOMAINS
1997
In papers [1,2] there were consider different assumptions for averaging methods along the vertical coordinate.These methods were applied for the mathematical simulation of the mass transfer process in multilayered underground systems. A specific feature of these problems is that it is necessity to solve the 3‐D initial‐boundary‐value problems for parabolic type partial differential equations of second order with piece‐wise parameters in multilayer domain.Therefore here an effective finite‐difference method for solving a problem of the above type is developed.This method may be considered as a generalization of the method of finite volumes [3] for the layered systems. In the case of constant…
Scheduled Relaxation Jacobi method: improvements and applications
2016
Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficien…
Analysis of the finite difference time domain technique to solve the Schrödinger equation for quantum devices
2004
An extension of the finite difference time domain is applied to solve the Schrödinger equation. A systematic analysis of stability and convergence of this technique is carried out in this article. The numerical scheme used to solve the Schrödinger equation differs from the scheme found in electromagnetics. Also, the unit cell employed to model quantum devices is different from the Yee cell used by the electrical engineering community. A bound for the time step is derived to ensure stability. Several numerical experiments in quantum structures demonstrate the accuracy of a second order, comparable to the analysis of electromagnetic devices with the Yee cell. a!Electronic mail: Antonio.Sorian…
Finite difference time domain simulation of soil ionization in grounding systems under lightning surge conditions
2004
This paper proposes a Maxwell’s equations finite difference time domain (FDTD) approach for electromagnetic transients in ground electrodes in order to take into account the non linear effects due to soil ionization. A time variable soil resistivity method is used in order to simulate the soil breakdown, without the formulation of an initial hypothesis about the geometrical shape of the ionized zone around the electrodes. The model has been validated by comparing the computed results with available data found in technical literature referred to concentrated earths. Some application examples referred to complex grounding systems are reported to show the computational capability of the propos…
The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors
2015
Abstract In a non-local fractional-order model of thermal energy transport recently introduced by the authors, it is assumed that local and non-local contributions coexist at a given observation scale: while the first is described by the classical Fourier transport law, the second involves couples of adjacent and non-adjacent elementary volumes, and is taken as proportional to the product of the masses of the interacting volumes and their relative temperature, through a material-dependent, distance-decaying power-law function. As a result, a fractional-order heat conduction equation is derived. This paper presents a pertinent finite element method for the solution of the proposed fractional…