Novel WENO-format high-precision fractional derivative approximation method

A fractional derivative, high-precision technology, applied in electrical digital data processing, complex mathematical operations, CAD numerical modeling, etc., can solve problems such as complex and cumbersome calculation processes

Pending Publication Date: 2021-02-02
NANJING UNIV OF AERONAUTICS & ASTRONAUTICS
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  • Application Information

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Problems solved by technology

However, in the implementation process of WENO6 format, the calculation process of lin...

Method used

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  • Novel WENO-format high-precision fractional derivative approximation method
  • Novel WENO-format high-precision fractional derivative approximation method
  • Novel WENO-format high-precision fractional derivative approximation method

Examples

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Embodiment 1

[0125] Embodiment 1: The following fractional differential equation is given

[0126]

[0127] The initial conditions are:

[0128]

[0129] The boundary conditions are:

[0130] u(-L,t)=1, u(L,t)=0; (19)

[0131] in V=0.5 and L=1 or 10.

[0132] exist Figure 1-12 In the numerical simulation of , take D=0.02, L=1, h=2 / N, grid points N=100, time step τ=0.4h 2 , Figure 1-4 , Figure 5-8 , Figure 9-12 Numerical solutions of fractional derivatives of different orders α=1.2, 1.4, 1.6, 1.8 at T=0.001 are given respectively, and it can be noticed that the sharp transition is preserved for different α numerical simulations.

[0133] exist Figure 13-15 In the numerical simulation of , take D=0.2, L=10, h=20 / N, grid points N=200, time step τ=0.4h 2 , Figure 13-15 Numerical solutions of fractional derivative α=1.8 at T=1,2,3 are given.

[0134] exist Figure 16-18 In the numerical simulation of , take D=0.2, L=10, h=20 / N, grid points N=200, time step τ=0.3h 1.8 ...

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Abstract

The invention discloses a novel WENO-format high-precision fractional derivative approximation method. The method comprises the steps: decomposing a Caputo fractional derivative in a fractional differential equation into a classical second derivative and a weak singular integral in a Cartesian coordinate system, discretizing the classical second derivative by using a novel WENO format, and solvingthe weak singular integral by using Gauss-Jacobi quadra-ture; for a time derivative in an equation, using a three-order TVD Runge-Kutta discrete formula to discretize a semi-discrete finite difference format into a space-time full-discrete finite difference format, wherein the space-time full-discrete finite difference format is an iterative formula about a time layer, and an initial state valueis known; and according to the space-time full-discrete finite difference format, solving an approximate value of a next time layer through an iterative formula, and sequentially acquiring a numericalsimulation value in a calculation region at the end moment. The method can achieve six-order precision in the smooth area of a solution, and can keep the property of basically no oscillation in a discontinuous strong interruption area.

Description

technical field [0001] The invention belongs to the technical field of computational fluid dynamics engineering, and in particular relates to a novel WENO format high-precision fractional derivative approximation method. Background technique [0002] High-precision finite-difference and finite-volume WENO schemes are used to solve piecewise smooth solutions with discontinuities. Liu, Osher and Chan et al. first proposed the first third-order finite volume WENO scheme in 1994, which improved the utilization of calculation results and improved the r-order precision ENO scheme to r+1-order numerical precision. In 1996, Jiang and Shu further improved the construction strategy of the high-order finite-difference WENO scheme, so that the numerical accuracy can be increased to the 2r-1 order numerical accuracy, and a new construction framework for smoothing factors and nonlinear weights was designed. [0003] In the implementation process of the classic WENO format, the linear wei...

Claims

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Application Information

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IPC IPC(8): G06F17/13G06F30/23G06F30/28G06F113/08G06F119/14G06F111/10
CPCG06F17/13G06F30/23G06F30/28G06F2111/10G06F2113/08G06F2119/14
Inventor 张燕朱君
Owner NANJING UNIV OF AERONAUTICS & ASTRONAUTICS
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