A High-Order Difference Scheme for the Space and Time Fractional Bloch–Torrey Equation

2018 ◽  
Vol 18 (1) ◽  
pp. 147-164 ◽  
Author(s):  
Yun Zhu ◽  
Zhi-Zhong Sun
Keyword(s):  
Fractional Derivative ◽  
Difference Scheme ◽  
Dimensional Space ◽  
High Order ◽  
Stability And Convergence ◽  
Difference Operators ◽  
Riesz Fractional Derivative ◽  
Central Difference ◽  
Order Difference ◽  
Space And Time

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.

A High-Order Difference Scheme for the Generalized Cattaneo Equation

2012 ◽  
Vol 2 (2) ◽  
pp. 170-184 ◽  
Author(s):  
Seak-Weng Vong ◽  
Hong-Kui Pang ◽  
Xiao-Qing Jin
Keyword(s):  
Difference Scheme ◽  
Fractional Part ◽  
High Order ◽  
Stability And Convergence ◽  
Compact Difference Scheme ◽  
Order Difference ◽  
Compact Difference ◽  
Fractional Cattaneo Equation ◽  
The Stability ◽  
High Order Finite Difference

AbstractA high-order finite difference scheme for the fractional Cattaneo equation is investigated. The L1 approximation is invoked for the time fractional part, and a compact difference scheme is applied to approximate the second-order space derivative. The stability and convergence rate are discussed in the maximum norm by the energy method. Numerical examples are provided to verify the effectiveness and accuracy of the proposed difference scheme.

scholarly journals High-Order Compact Difference Scheme and Multigrid Method for Solving the 2D Elliptic Problems

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Yan Wang ◽  
Yongbin Ge
Keyword(s):  
Difference Scheme ◽  
Multigrid Method ◽  
Elliptic Problems ◽  
High Order ◽  
Compact Difference Scheme ◽  
Central Difference ◽  
Present Scheme ◽  
Compact Difference ◽  
Compact Approximations ◽  
Error Terms

A high-order compact difference scheme for solving the two-dimensional (2D) elliptic problems is proposed by including compact approximations to the leading truncation error terms of the central difference scheme. A multigrid method is employed to overcome the difficulties caused by conventional iterative methods when they are used to solve the linear algebraic system arising from the high-order compact scheme. Numerical experiments are conducted to test the accuracy and efficiency of the present method. The computed results indicate that the present scheme achieves the fourth-order accuracy and the effect of the multigrid method for accelerating the convergence speed is significant.

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

2001 ◽  
Vol 1 (4) ◽  
pp. 398-414 ◽  
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.

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

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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