WITHDRAWN: High-order numerical methods for the Riesz space fractional advection–dispersion equations

Feng, Libo; Zhuang, Pinghui; Liu, F.; Turner, Ian; Li, J.

doi:10.1016/j.camwa.2016.01.015

Cited by 28 publications

(11 citation statements)

References 26 publications

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“…Theorem 1 allows us to use the Riemann-Liouville fractional derivative definition for the formulation of the problem. The weighted shifted Grünwald-Letnikov derivative formula for approximating the two-sided fractional derivative is derived in References [46,48] for space fractional derivative and Crank-Nicolson scheme for time are used.…”

Section: Numerical Approximation For One Dimensional Two-sided Convection-diffusion Problem With Source Termmentioning

confidence: 99%

Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

Anley

Zheng

2020

Mathematics

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In this paper, we have considered a numerical difference approximation for solving two-dimensional Riesz space fractional convection-diffusion problem with source term over a finite domain. The convection and diffusion equation can depend on both spatial and temporal variables. Crank-Nicolson scheme for time combined with weighted and shifted Grünwald-Letnikov difference operator for space are implemented to get second order convergence both in space and time. Unconditional stability and convergence order analysis of the scheme are explained theoretically and experimentally. The numerical tests are indicated that the Crank-Nicolson scheme with weighted shifted Grünwald-Letnikov approximations are effective numerical methods for two dimensional two-sided space fractional convection-diffusion equation.

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Section: Numerical Approximation For One Dimensional Two-sided Convection-diffusion Problem With Source Termmentioning

confidence: 99%

Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

Anley

Zheng

2020

Mathematics

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“….. If = 0.5, then (6) becomes the Crank-Nicolson (CN) scheme [46] and the convergence rate is (Δ 2 )+ (ℎ 2 ). If = 1, then (6) becomes a fully implicit scheme with the convergence rate, (Δ ) + (ℎ 2 ).…”

Section: Numerical Solutionmentioning

confidence: 99%

Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Jeong

Lee

et al. 2019

Mathematical Problems in Engineering

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In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

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“…In this section, we provide the stability and convergence analyses for the finite difference method for the space fractional equation. We first present some lemmas.Lemma ([35]) Let

A

be an (m − 1)‐order positive define matrix; then for any parameter θ ≥ 0, the following two inequalities

‖ {((), I + θ scriptA)}^{- 1} ‖ \leq 1

and

‖ {((), I + θ scriptA)}^{- 1} (I - θ A) ‖ \leq 1

hold . Lemma ([36]) A real matrix D of order n is positive definite if and only if its symmetric part

H = \frac{D + D^{T}}{2}

is positive definite; H is positive definite if and only if the eigenvalues of H are positive .…”

Section: Convergence and Stability Analysismentioning

confidence: 99%

Novel numerical techniques for the finite moment log stable computational model for European call option

Liu

Chen

et al. 2020

Numerical Methods Partial

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Option pricing models are often used to describe the dynamic characteristics of prices in financial markets. Unlike the classical Black-Scholes (BS) model, the finite moment log stable (FMLS) model can explain large movements of prices during small time steps. In the FMLS, the second-order spatial derivative of the BS model is replaced by a fractional operator of order which generates an-stable Lévy process. In this paper, we consider the finite difference method to approximate the FMLS model. We present two numerical schemes for this approximation: the implicit numerical scheme and the Crank-Nicolson scheme. We carry out convergence and stability analyses for the proposed schemes. Since the fractional operator routinely generates dense matrices which often require high computational cost and storage memory, we explore three methods for solving the approximation schemes: the Gaussian elimination method, the bi-conjugate gradient stabilized method (Bi-CGSTAB) and the fast Bi-CGSTAB (FBi-CGSTAB) in order to compare the cost of calculations. Finally, two numerical examples with exact solutions are presented where we also use extrapolation techniques to achieve higher-order convergence. The results suggest that the proposed schemes are unconditionally stable and convergent, and the FMLS model is useful for pricing options.

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WITHDRAWN: High-order numerical methods for the Riesz space fractional advection–dispersion equations

Cited by 28 publications

References 26 publications

Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Novel numerical techniques for the finite moment log stable computational model for European call option

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