Analysis Of Finite Difference Schemes Research Articles

Error analysis of finite difference scheme for American option pricing under regime-switching with jumps

This paper mainly focuses on evaluating American options under regime-switching jump-diffusion models (Merton’s and Kou’s models). An efficient numerical method is designed for the concerned problems. The problem of American option pricing under regime-switching jump-diffusion models can be described as a free-boundary problem or a complementarity problem with integral and differential terms on an unbounded domain. By analyzing the relation of optimal exercise boundaries among several options, we truncate the solving domain of regime-switching jump-diffusion options, and present reasonable boundary conditions. For the integral terms of the truncated model, a composite trapezoidal formula is applied, which guarantees that the integral discretized matrix is a Toeplitz matrix. Meanwhile, a finite difference scheme is proposed for the resulting system, which leads to a linear complementary problem (LCP) with a unique solution. Moreover, we also prove the stability, monotonicity, and consistency of the discretization scheme and estimate the convergence order. In consideration of the characteristics of the discrete matrix, a projection and contraction method is suggested to solve the discretized LCP. Numerical experiments are carried out to verify the efficiency of the proposed scheme.

Numerical analysis of finite difference schemes arising from time-memory partial integro-differential equations

This paper investigates the partial integro-differential equation of memory type numerically. The differential operator is discretized based on θ-finite difference schemes, while the integral operator is approximated using Simpson's rule. The mesh points of an integral part are adapted to coincide with the nodes of the computational domain using the Heaviside function. The stability of the proposed numerical methods is established based on Gerschgoren's theorems. Also, its consistency is investigated to guarantee the numerical solutions' convergence. Several examples are provided to discuss the efficiency of the used finite difference schemes and compare them with previous studies.

Analysis of finite difference schemes for Volterra integro-differential equations involving arbitrary order derivatives

Analysis of finite difference schemes for Volterra integro-differential equations involving arbitrary order derivatives

Finite Difference formulation of any lattice Boltzmann scheme

Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of consistency with respect to the target equations and to provide a precise notion of stability. In order to alleviate these shortages and introduce a rigorous framework, we demonstrate that any lattice Boltzmann scheme can be rewritten as a corresponding multi-step Finite Difference scheme on the conserved variables. This is achieved by devising a suitable formalism based on operators, commutative algebra and polynomials. Therefore, the notion of consistency of the corresponding Finite Difference scheme allows to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequently-used von Neumann-like stability analysis for lattice Boltzmann schemes entirely corresponds to the von Neumann stability analysis of their Finite Difference counterpart. More generally, the usual tools for the analysis of Finite Difference schemes are now readily available to study lattice Boltzmann schemes. Their relevance is verified by means of numerical illustrations.

Analysis of finite difference schemes for a fourth-order strongly damped nonlinear wave equations

In this paper, we propose two finite difference schemes to solve a class of a fourth-order strongly damped nonlinear wave equation in two dimensions. We prove some a priori bounds and establish the second order convergence in L∞−norm for the difference solutions. We also discuss the stability and the unique solvability of the two proposed difference schemes. Finally, we provide numerical examples to validate the order of convergence, unconditional stability and the nonincreasing discrete energy in time for the two introduced schemes.

Stability analysis of finite difference schemes for two-dimensional hyperbolic equations using Fourier transforms

Stability analysis of finite difference schemes for two-dimensional hyperbolic equations using Fourier transforms

Unsteady magnetohydrodynamic casson nanofluid flow through a moving cylinder with brownian and thermophoresis effects

The aim of this study is to analysis of finite difference scheme of unsteady MHD flow of Casson nano-fluid attribute of Brownian motion and thermophoresis through a moving cylinder. The governing model for the flow is metamorphosed into non-dimensional impetus, strength and mass-diffusion equations and evolved numerically by employing explicit finite difference fetch with the aid of a computer programming language Compact visual FORTRAN 6.6a. In order to optimize the strait parameters and exactness of the strait, the stability and convergence test have sustained. It is clear that with primary boundary postulates, U= V= T= C= 0, and small difference time Δt=0.0005, ΔX=0.202, and ΔR= 0.251, the strait has converged for Prandtl number, Pr ≥ 0.02 and Lewis number, Le ≥ 0.018. The acquired results of this study are discussed for several values of natural parameters viz. Prandtl number, Casson fluid parameter, Lewis number, magnetic parameter, Brownian motion and thermophoresis number on the impetus, strength, mass-diffusion, skin friction, Nusselt number by means of several time steps. Moreover, the graphical representations of the solution are shown by conducting tecplot 9.0.

Stability analysis of finite difference schemes for an advection diffusion equation

The paper studies stability analysis for two standard finite difference schemes FTBSCS (forward time backward space and centered space) and FTCS (forward time and centered space). One-dimensional advection diffusion equation is solved by using the schemes with appropriate initial and boundary conditions. Numerical experiments are performed to verify the stability results obtained in this study. It is found that FTCS scheme gives better point-wise solutions than FTBSCS in terms of time step selection.Bangladesh J. Sci. Res. 29(2): 143-151, December-2016

Spectral analysis of finite difference schemes for convection diffusion equation

The paper presents numerical analysis of finite difference schemes for solving the linear convection-diffusion equation using a full domain spectral analysis method illustrated in Sengupta et al. (2003) [7]. Different numerical schemes ranging from simple central and upwind difference schemes to high accuracy schemes like compact and combined compact difference schemes are analyzed for their accuracy. Optimal values of simulation parameters are proposed for the analyzed schemes with a view to obtain accurate solutions.

Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes

In this paper, numerical analysis of finite difference schemes for partial integro-differential models related to European and American option pricing problems under a wide class of Lévy models is studied. Apart from computational and accuracy issues, qualitative properties such as positivity are treated. Consistency of the proposed numerical scheme and stability in the von Neumann sense are included. Gauss–Laguerre quadrature formula is used for the discretization of the integral part. Numerical examples illustrating the potential advantages of the presented results are included.

On Stability Analysis of Finite-Difference Schemes for Some Parabolic Problems with Nonlocal Boundary Conditions

In this article, one-dimensional parabolic and pseudo-parabolic equations with nonlocal boundary conditions are approximated by the implicit Euler finite-difference scheme. For a parabolic problem, the stability analysis is done in the weak H −1 type norm, which enables us to generalize results obtained in stronger norms. In the case of a pseudo-parabolic problem, the stability analysis is done in the discrete analog of the norm. It is shown that a solution of the proposed finite-discrete scheme satisfies stronger stability estimates than a discrete solution of the parabolic problem. Results of numerical experiments are presented.

The von Neumann analysis and modified equation approach for finite difference schemes

The von Neumann (discrete Fourier) analysis and modified equation technique have been proven to be two effective tools in the design and analysis of finite difference schemes for linear and nonlinear problems. The former has merits of simplicity and intuition in practical applications, but only restricted to problems of linear equations with constant coefficients and periodic boundary conditions. The later PDE approach has more extensive potential to nonlinear problems and error analysis despite its kind of relative complexity. The dissipation and dispersion properties can be observed directly from the PDE point of view: Even-order terms supply dissipation and odd-order terms reflect dispersion. In this paper we will show rigorously their full equivalence via the construction of modified equation of two-level finite difference schemes around any wave number only in terms of the amplification factor used in the von Neumann analysis. Such a conclusion fills in the gap between these two approaches in literatures.

Stability analysis of finite difference schemes revisited: A study of decoupled solution strategies for coupled multifield problems

Many problems in engineering, physics or other disciplines require an integrated treatment of coupled heterogeneous fields. The mathematical models of such problems commonly consist of a coupled PDE system. Considerable progress has been made in the development of schemes to solve such systems resulting in a wide spectrum of monolithic and decoupled numerical solution approaches. In particular, the flexibility offered by the decoupled strategies in choosing appropriate solution methods and adjustable time-step sizes has motivated the elaboration of these schemes. However, the way of decoupling can influence the stability of the resulting solution algorithm and, hence, urges a comprehensive stability analysis.The purpose of this contribution is to elucidate the salient points in the stability analysis of the numerical solution schemes. To this end, we introduce a general framework for the stability analysis of solution strategies and provide an easy-to-use procedure to find the necessary stability condition. The procedure is tested against decoupled solution strategies for volume- and surface-coupled problems, namely, thermoelastodynamics, porous media dynamics, and fluid– and structure–structure interaction. In this context, the influence of explicit and implicit predictors and the sequence of the time integration on the stability condition is carefully examined. Copyright © 2013 John Wiley & Sons, Ltd.

Uniform Convergence Analysis of Finite Difference Scheme for Singularly Perturbed Delay Differential Equation on an Adaptively Generated Grid

Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper, we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function. It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter. Numerical experiments illustrate in practice the result of convergence proved theoretically. where 0 0,∀x∈ . Such a problem is sometimes addressed as two-parameter problem. The argument for small delay problems

Asymptotic Analysis of Lattice Boltzmann Boundary Conditions

In this article, we use a general method for the analysis of finite difference schemes to investigate lattice Boltzmann algorithms for Navier–Stokes problems with Dirichlet boundary conditions. Several link based boundary conditions for commonly used lattice Boltzmann BGK models are considered. With our method, the accuracy of the algorithms can be exactly predicted. Moreover, the analytical results can be used to construct new algorithms which is demonstrated with a corrected bounce back rule that requires only local evaluations but still yields second order accuracy for the velocity. The analysis is applicable to general geometries and instationary flows

Numerical Experiments Aimed at the Comparison of Two Finite-Difference Schemes for the Equations of Motion in a Discrete Model of Hydrodynamics of the Black Sea

By using the box method and the method of indefinite coefficients, we perform the analysis of finite-difference schemes with two invariants for the equations of motion in the barotropic approximation. We establish a functional dependence, which makes it possible to construct difference schemes with the properties of antisymmetry of the solution and conservation of energy and/or enstrophy. Two approximations of nonlinear terms in the equations of motion are analyzed. The first of these approximations guarantees the conservation of two quadratic invariants for the barotropic divergent motion. In the second approximation, the same is true for the barotropic nondivergent motion. The results of prognostic experiments (20 yr of model time) demonstrate that the first scheme gives a more accurate quantitative description of the well-known physical features of large-scale circulation in the Black Sea.

Asymptotic analysis of finite difference methods

With this article, we want to advocate the use of asymptotic methods for the analysis of finite difference schemes. We present several examples to demonstrate the applicability of the approach. Advantages over the modified equation and truncation error analysis are pointed out.

Analysis of finite difference schemes for unsteady Navier-Stokes equations in vorticity formulation

In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible Navier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted into local vorticity boundary conditions. Thom's formula, Wilkes' formula, or other local formulas in the earlier literature can be used in the second order method; while high order formulas, such as Briley's formula, can be used in the fourth order compact difference scheme proposed by E and Liu. The stability analysis of these long-stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula. The main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function. We choose to analyze the second order scheme with Wilkes' formula in detail. In this case, we can avoid the complicated technique necessitated by the Strang-type high order expansions. As a consequence, our analysis results in almost optimal regularity assumption for the exact solution. The above methodology is very general. We also give a detailed analysis for the fourth order scheme using a 1-D Stokes model.

The Finite Difference Scheme for 3d Mathematical Modeling of a Wood Drying Process

AbstractThis work discusses issues on the design and analysis of finite difference schemes for 3D modeling the process of moisture motion in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for Robin boundary conditions. The influence of boundary conditions is investigated, and results of numerical experiments are presented.

The remainder-effect analysis of finite difference schemes and the applications

In the present paper, two contents are enclosed. First. the Fourier analysis approac.h of the dispersion relation and group velocio, effect of .finite difference schemes is discussed, the defects of the approach is pointed out and the correction is made; Second, a new systematic analysis method-remaider-effect analysis (abbr. REAM) is proposed by means of the modified partial differential equations (abbr. MPDE) of finite difference schemes. The analysis is based on the synthetical stud), of the rational dispersion-and dissipation relations of finite difference schemes. And the method clearly possesses constructivity.