Hermite Spectral Methods with a Time-Dependent Scaling for Parabolic Equations in Unbounded Domains
Hermite Spectral Methods with a Time-Dependent Scaling for Parabolic Equations in Unbounded Domains
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DOI：
10.1137/s0036142903421278
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
He-ping Ma;Weiwei Sun;T. Tang
中科院分区：
文献类型：
--
作者：
He-ping Ma;Weiwei Sun;T. Tang
研究了无界区域上线性扩散方程和非线性对流扩散方程的Hermite谱方法。当解域无界时，扩散算子不再有紧预解式，这使得Hermite谱方法不稳定。为了克服这个困难，一个时间相关的比例因子中使用的Hermite展开，这产生了一个积极的双线性形式。因此，稳定性和谱收敛可以建立这种方法。本方法在由Funaro和Kavian [Math. Comp.，57（1991），pp. 597- 619]。然而，由于不需要坐标变换，因此本方法更有效并且更容易实现。事实上，与时间相关的缩放所得到的离散化系统是相同的形式与经典的（简单的，但不稳定的）埃尔米特谱方法。数值实验证明了该方法的稳定性和收敛性。
Hermite spectral methods are investigated for linear diffusion equations and nonlinear convection-diffusion equations in unbounded domains. When the solution domain is unbounded, the diffusion operator no longer has a compact resolvent, which makes the Hermite spectral methods unstable. To overcome this difficulty, a time-dependent scaling factor is employed in the Hermite expansions, which yields a positive bilinear form. As a consequence, stability and spectral convergence can be established for this approach. The present method plays a similar role in the stability of the {similarity transformation} technique proposed by Funaro and Kavian [Math. Comp., 57 (1991), pp. 597--619]. However, since coordinate transformations are not required, the present approach is more efficient and is easier to implement. In fact, with the time-dependent scaling the resulting discretization system is of the same form as that associated with the classical (straightforward but unstable) Hermite spectral method. Numerical experiments are carried out to support the theoretical stability and convergence results.