An analysis for a high-order difference scheme for numerical solution toutt =A(x, t)uxx +F(x, t, u, ut,ux)

AbstractIn this paper, a high-order difference scheme is proposed for an one-dimensional space and time fractional Bloch–Torrey equation. A third-order accurate formula, based on the weighted and shifted Grünwald–Letnikov difference operators, is used to approximate the Caputo fractional derivative in temporal direction. For the discretization of the spatial Riesz fractional derivative, we approximate the weighed values of the Riesz fractional derivative at three points by the fractional central difference operator. The unique solvability, unconditional stability and convergence of the scheme are rigorously proved by the discrete energy method. The convergence order is 3 in time and 4 in space in {L_{1}(L_{2})}-norm. Two numerical examples are implemented to testify the accuracy of the numerical solution and the efficiency of the difference scheme.

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A High-Order Difference Scheme for a Nonlocal Boundary-Value Problem for the Heat Equation

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2001-0026 ◽

2001 ◽

Vol 1 (4) ◽

pp. 398-414 ◽

Cited By ~ 9

Author(s):

Zhi-Zhong Sun

Keyword(s):

Boundary Value Problem ◽

Difference Scheme ◽

Boundary Value ◽

Nonlocal Boundary ◽

High Order ◽

Order Difference ◽

Local Boundary ◽

Non Local ◽

The Difference ◽

Boundary Equations

Abstract This paper is concerned with a high order difference scheme for a non- local boundary-value problem of parabolic equation. The integrals in the boundary equations are approximated by the composite Simpson rule. The unconditional solv- ability and L_∞ convergence of the difference scheme is proved by the energy method. The convergence rate of the difference scheme is second order in time and fourth order in space. Some numerical examples are provided to illustrate the convergence.

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Design of High-Order Difference Scheme and Analysis of Solution Characteristics—Part I: General Formulation of High-Order Difference Schemes and Analysis of Convective Stability

Numerical Heat Transfer Part B Fundamentals ◽

10.1080/10407790701372769 ◽

2007 ◽

Vol 52 (3) ◽

pp. 231-254 ◽

Cited By ~ 3

Author(s):

W. W. Jin ◽

W. Q. Tao

Keyword(s):

Difference Scheme ◽

High Order ◽

Difference Schemes ◽

Convective Stability ◽

Order Difference

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Valuation of the American put option as a free boundary problem through a high-order difference scheme

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2020-0252 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Murat Sari ◽

Seda Gulen

Keyword(s):

Free Boundary ◽

Difference Scheme ◽

Boundary Problem ◽

Free Boundary Problem ◽

American Options ◽

High Order ◽

Order Difference ◽

Put Option ◽

A Free Boundary Problem ◽

American Put

Abstract Valuation of the American options encountered commonly in finance is quite difficult due to the possibility of early exercise alternatives. Since an exact solution for the American options does not exist, effective numerical methods are needed to understand the behavior of option pricing models. Therefore, in this paper, a new approach based on a high-order difference scheme is proposed to discuss the valuation of an American put option as a free boundary problem. Using a front-fixing approach that transforms the unknown free boundary (optimal stopping) into a fixed one, a sixth-order finite difference scheme (FD6) in space and a third-order strong-stability preserving Runge–Kutta (SSPRK3) in time are applied to the model converted to a nonlinear partial differential equation. The computed results revealed that the combined method is seen to attempt to pull up the capacity of the algorithm to achieve higher accuracy. It is seen that the quantitative and qualitative results produced by the method proposed with minimal computational effort are sufficiently accurate and meaningful. Therefore, this article provides some new insights about the physical characteristics of financial problems and such realistic phenomena.

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