ALTERNATING DIRECTION METHOD FOR A TWO-DIMENSIONAL PARABOLIC EQUATION WITH A NONLOCAL BOUNDARY CONDITION

The present paper deals with an alternating direction implicit method for a two dimensional parabolic equation in a rectangle domain with a nonlocal boundary condition in one direction. Sufficient conditions of stability for Peaceman‐Rachford method are established. Results of some numerical experiments are presented.

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Pedestrian Walking Behavior Revealed through a Random Walk Model

Discrete Dynamics in Nature and Society ◽

10.1155/2012/405907 ◽

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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Combination of the Improved Diffraction Nonlocal Boundary Condition and Three-Dimensional Wide-Angle Parabolic Equation Decomposition Model for Predicting Radio Wave Propagation

International Journal of Antennas and Propagation ◽

10.1155/2017/2728380 ◽

2017 ◽

Vol 2017 ◽

pp. 1-10

Author(s):

Ruidong Wang ◽

Guizhen Lu ◽

Rongshu Zhang ◽

Weizhang Xu

Keyword(s):

Boundary Condition ◽

Wave Propagation ◽

Parabolic Equation ◽

Nonlocal Boundary Condition ◽

Nonlocal Boundary ◽

Three Dimensional ◽

Wide Angle ◽

Computational Accuracy ◽

Decomposition Model ◽

Convolution Integrals

Diffraction nonlocal boundary condition (BC) is one kind of the transparent boundary condition which is used in the finite-difference (FD) parabolic equation (PE). The greatest advantage of the diffraction nonlocal boundary condition is that it can absorb the wave completely by using one layer of grid. However, the speed of computation is low because of the time-consuming spatial convolution integrals. To solve this problem, we introduce the recursive convolution (RC) with vector fitting (VF) method to accelerate the computational speed. Through combining the diffraction nonlocal boundary with RC, we achieve the improved diffraction nonlocal BC. Then we propose a wide-angle three-dimensional parabolic equation (WA-3DPE) decomposition algorithm in which the improved diffraction nonlocal BC is applied and we utilize it to predict the wave propagation problems in the complex environment. Numeric computation and measurement results demonstrate the computational accuracy and speed of the WA-3DPE decomposition model with the improved diffraction nonlocal BC.

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