scholarly journals A Mixed Element Algorithm Based on the Modified L1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model

2021 ◽  
Vol 5 (4) ◽  
pp. 274
Author(s):  
Jinfeng Wang ◽  
Baoli Yin ◽  
Yang Liu ◽  
Hong Li ◽  
Zhichao Fang
Keyword(s):  
Fourth Order ◽  
Mixed Finite Element ◽  
Calculated Data ◽  
Coupled System ◽  
Wave Model ◽  
Fractional Diffusion ◽  
Second Order ◽  
Diffusion Wave ◽  
Nicolson Scheme ◽  
Crank Nicolson

In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified L1-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal L2 error estimates are performed and the feasibility is validated by the calculated data.

scholarly journals Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay

2021 ◽  
Vol 57 ◽  
pp. 128-141
Author(s):  
M. Ibrahim ◽  
V.G. Pimenov
Keyword(s):  
Piecewise Linear ◽  
Linear Interpolation ◽  
Fractional Diffusion ◽  
Dimensional System ◽  
Two Dimensional ◽  
Piecewise Linear Interpolation ◽  
Functional Delay ◽  
General Difference ◽  
Nicolson Scheme ◽  
Crank Nicolson

A two-dimensional in space fractional diffusion equation with functional delay of a general form is considered. For this problem, the Crank-Nicolson method is constructed, based on shifted Grunwald-Letnikov formulas for approximating fractional derivatives with respect to each spatial variable and using piecewise linear interpolation of discrete history with continuation extrapolation to take into account the delay effect. The Douglas scheme is used to reduce the emerging high-dimensional system to tridiagonal systems. The residual of the method is investigated. To obtain the order of the method, we reduce the systems to constructions of the general difference scheme with heredity. A theorem on the second order of convergence of the method in time and space steps is proved. The results of numerical experiments are presented.

scholarly journals A New Alternating Segment Crank-Nicolson Scheme for the Fourth-Order Parabolic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Ge-yang Guo ◽  
Bo Liu
Keyword(s):  
Parabolic Equation ◽  
Fourth Order ◽  
Unconditional Stability ◽  
Truncation Errors ◽  
Order Parabolic Equation ◽  
Intrinsic Parallelism ◽  
Fourth Order Parabolic Equation ◽  
The Stability ◽  
Nicolson Scheme ◽  
Crank Nicolson

A group of asymmetric difference schemes to approach the fourth-order parabolic equation is given. According to these schemes and the Crank-Nicolson scheme, an alternating segment Crank-Nicolson scheme with intrinsic parallelism is constructed. The truncation errors and the stability are discussed. Numerical simulations show that this new scheme has unconditional stability and high accuracy and convergency, and it is in preference to the implicit scheme method.

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