Some high order difference schemes for the space and time fractional Bloch–Torrey equations

2016 ◽  
Vol 281 ◽  
pp. 356-380 ◽  
Author(s):  
Hong Sun ◽  
Zhi-zhong Sun ◽  
Guang-hua Gao
Keyword(s):  
High Order ◽  
Difference Schemes ◽  
Order Difference ◽  
Space And Time

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.

scholarly journals HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs

2015 ◽  
Vol 18 (05) ◽  
pp. 1550031 ◽  
Author(s):  
ANDREY ITKIN
Keyword(s):  
Finite Difference ◽  
Order Approximation ◽  
High Order ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Option Prices ◽  
First Order ◽  
Space And Time ◽  
Discrete Dividends ◽  
First Order Approximation

This paper is dedicated to the construction of high order (in both space and time) finite-difference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations are consistent. This approach is partly inspired by Andreassen & Huge (2011) who reported a pair of consistent finite-difference schemes of first-order approximation in time for an uncorrelated local stochastic volatility (LSV) model. We extend their approach by constructing schemes that are second-order in both space and time and that apply to models with jumps and discrete dividends. Taking correlation into account in our approach is also not an issue.

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