The artificial boundary conditions for incompressible materials on an unbounded domain

1997 ◽  
Vol 77 (3) ◽  
pp. 347-363 ◽  
Author(s):  
H. Han ◽  
W. Bao
Keyword(s):  
Boundary Conditions ◽  
Unbounded Domain ◽  
Artificial Boundary ◽  
Artificial Boundary Conditions ◽  
Incompressible Materials

A FDM for Burgers’ Equation on Unbounded Domains Using High-Order Artificial Boundary Conditions

2017 ◽  
Vol 865 ◽  
pp. 233-238
Author(s):  
Quan Zheng ◽  
Yu Feng Liu
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Unbounded Domain ◽  
Burgers Equation ◽  
High Order ◽  
Computational Domain ◽  
Artificial Boundary ◽  
Artificial Boundary Conditions ◽  
Second Order Convergence ◽  
The Stability

Burgers’ equation on an unbounded domain is an important mathematical model to treat with some external problems of fluid materials. In this paper, we study a FDM of Burgers’ equation using high-order artificial boundary conditions on the unbounded domain. First, the original problem is converted into the heat equation on an unbounded domain by Hopf-Cole transformation. Thus the difficulty of nonlinearity of Burgers’ equation is overcome. Second, high-order artificial boundary conditions are given by using Padé approximation and Laplace transformation. And the conditions confine the heat equation onto a bounded computational domain. Third, we prove the solutions of the resulting heat equation and Burgers’ equation are both stable. Fourth, we establish the FDM for Burgers’ equation on the bounded computational domain. Finally, a numerical example demonstrates the stability, the effectiveness and the second-order convergence of the proposed method.

Numerical Solution of the Time-Fractional Sub-Diffusion Equation on an Unbounded Domain in Two-Dimensional Space

2017 ◽  
Vol 7 (3) ◽  
pp. 439-454 ◽  
Author(s):  
Hongwei Li ◽  
Xiaonan Wu ◽  
Jiwei Zhang
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Unbounded Domain ◽  
Dimensional Space ◽  
Fourier Series Expansion ◽  
Computational Domain ◽  
Two Dimensional ◽  
Artificial Boundary ◽  
Artificial Boundary Conditions

AbstractThe numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space is considered, where a circular artificial boundary is introduced to divide the unbounded domain into a bounded computational domain and an unbounded exterior domain. The local artificial boundary conditions for the fractional sub-diffusion equation are designed on the circular artificial boundary by a joint Laplace transform and Fourier series expansion, and some auxiliary variables are introduced to circumvent high-order derivatives in the artificial boundary conditions. The original problem defined on the unbounded domain is thus reduced to an initial boundary value problem on a bounded computational domain. A finite difference and L1 approximation are applied for the space variables and the Caputo time-fractional derivative, respectively. Two numerical examples demonstrate the performance of the proposed method.

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