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Published in:
2011 | OriginalPaper | Chapter
Introduction
Authors : Weizhang Huang, Robert D. Russell
Published in: Adaptive Moving Mesh Methods
Publisher: Springer New York
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In this first chapter, we introduce the basic principles of adaptivity and moving mesh methods for solving partial differential equations in one spatial dimension. In particular, two adaptive methods are described and used to solve a simple model problem - an initial-boundary value problem consisting of Burgers’ equation
1.1
$$u_t={\varepsilon}{u}_{xx}-\left(\frac{u^2}{2}\right)_x, \ \ x \in(0,1), t>0$$
subject to the boundary conditions
1.2
$$u(0,t)=u(1,t)=0$$
and initial condition
1.3
$$u(0,0)=\sin(2\pi x)+\frac{1}{2}\sin (\pi x)$$