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

2007 ◽  
Vol 52 (3) ◽  
pp. 231-254 ◽  
Author(s):  
W. W. Jin ◽  
W. Q. Tao
Keyword(s):  
Difference Scheme ◽  
High Order ◽  
Difference Schemes ◽  
Convective Stability ◽  
Order Difference

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 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.

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