scholarly journals Maximum Norm Error Estimates of ADI Methods for a Two-Dimensional Fractional Subdiffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Yuan-Ming Wang
Keyword(s):  
Finite Difference Methods ◽  
Maximum Norm ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Truncation Errors ◽  
Alternating Direction ◽  
Difference Methods ◽  
Subdiffusion Equation ◽  
Adi Methods ◽  
Norm Error

This paper is concerned with two alternating direction implicit (ADI) finite difference methods for solving a two-dimensional fractional subdiffusion equation. An explicit error estimate for each of the two methods is provided in the discrete maximum norm. It is shown that the methods have the same order as their truncation errors with respect to the discrete maximum norm. Numerical results are given to confirm the theoretical analysis results.

scholarly journals Alternating Direction Implicit (ADI) Methods for Solving Two-Dimensional Parabolic Interface Problems with Variable Coefficients

Computation ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 79
Author(s):  
Chuan Li ◽  
Guangqing Long ◽  
Yiquan Li ◽  
Shan Zhao
Keyword(s):  
Convergence Rates ◽  
Time Integration ◽  
Variable Coefficients ◽  
Interface Problems ◽  
Implicit Time ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Fully Implicit ◽  
Alternating Direction ◽  
Adi Methods

The matched interface and boundary method (MIB) and ghost fluid method (GFM) are two well-known methods for solving elliptic interface problems. Moreover, they can be coupled with efficient time advancing methods, such as the alternating direction implicit (ADI) methods, for solving time-dependent partial differential equations (PDEs) with interfaces. However, to our best knowledge, all existing interface ADI methods for solving parabolic interface problems concern only constant coefficient PDEs, and no efficient and accurate ADI method has been developed for variable coefficient PDEs. In this work, we propose to incorporate the MIB and GFM in the framework of the ADI methods for generalized methods to solve two-dimensional parabolic interface problems with variable coefficients. Various numerical tests are conducted to investigate the accuracy, efficiency, and stability of the proposed methods. Both the semi-implicit MIB-ADI and fully-implicit GFM-ADI methods can recover the accuracy reduction near interfaces while maintaining the ADI efficiency. In summary, the GFM-ADI is found to be more stable as a fully-implicit time integration method, while the MIB-ADI is found to be more accurate with higher spatial and temporal convergence rates.

Numerical studies of high-dimensional nonlinear viscous and nonviscous wave equations by using finite difference methods

2020 ◽  
Vol 11 (02) ◽  
pp. 2050008 ◽  
Author(s):  
Ding-Wen Deng ◽  
Zhu-An Wang
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Three Dimensional ◽  
Discrete Energy ◽  
Alternating Direction Implicit ◽  
Numerical Studies ◽  
Alternating Direction ◽  
Difference Methods

The numerical solutions of two-dimensional (2D) and three-dimensional (3D) nonlinear viscous and nonviscous wave equations via the unified alternating direction implicit (ADI) finite difference methods (FDMs) are obtained in this paper. By making use of the discrete energy method, it is proven that their numerical solutions converge to exact solutions with an order of two in both time and space with respect to [Formula: see text]-norm. Numerical results confirm that they are relatively accurate and high-resolution, and more successfully simulate the conservation of the energy for nonviscous equations, and the dissipation of the energy for viscous equation.

Three-Point Combined Compact Alternating Direction Implicit Difference Schemes for Two-Dimensional Time-Fractional Advection-Diffusion Equations

2015 ◽  
Vol 17 (2) ◽  
pp. 487-509 ◽  
Author(s):  
Guang-Hua Gao ◽  
Hai-Wei Sun
Keyword(s):  
Diffusion Equations ◽  
Two Dimensional ◽  
Analysis Method ◽  
Order Accuracy ◽  
Alternating Direction Implicit ◽  
Truncation Errors ◽  
Advection Diffusion ◽  
Alternating Direction ◽  
Implicit Difference Schemes ◽  
Adi Schemes

AbstractThis paper is devoted to the discussion of numerical methods for solving two-dimensional time-fractional advection-diffusion equations. Two different three-point combined compact alternating direction implicit (CC-ADI) schemes are proposed and then, the original schemes for solving the two-dimensional problems are divided into two separate one-dimensional cases. Local truncation errors are analyzed and the unconditional stabilities of the obtained schemes are investigated by Fourier analysis method. Numerical experiments show the effectiveness and the spatial higher-order accuracy of the proposed methods.

scholarly journals Pedestrian Walking Behavior Revealed through a Random Walk Model

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Hui Xiong ◽  
Liya Yao ◽  
Huachun Tan ◽  
Wuhong Wang
Keyword(s):  
Numerical Experiments ◽  
Flow Simulation ◽  
Two Dimensional ◽  
Pedestrian Flow ◽  
Alternating Direction Implicit Method ◽  
Walking Behavior ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Continuous Time Random Walks ◽  
High Order Compact Scheme

This paper applies method of continuous-time random walks for pedestrian flow simulation. In the model, pedestrians can walk forward or backward and turn left or right if there is no block. Velocities of pedestrian flow moving forward or diffusing are dominated by coefficients. The waiting time preceding each jump is assumed to follow an exponential distribution. To solve the model, a second-order two-dimensional partial differential equation, a high-order compact scheme with the alternating direction implicit method, is employed. In the numerical experiments, the walking domain of the first one is two-dimensional with two entrances and one exit, and that of the second one is two-dimensional with one entrance and one exit. The flows in both scenarios are one way. Numerical results show that the model can be used for pedestrian flow simulation.

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