Accurate numerical resolution of transients in initial-boundary value problems for the heat equation

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• 作者：Flyer ; Natasha ; Fornberg ; Bengt
• 关键词：PDE ; Initial-boundary value problem ; Corner singularity ; Heat equation ; Chebyshev method
• 刊名：Journal of Computational Physics
• 出版年：2003
• 出版时间：January 20, 2003
• 年：2003
• 卷：184
• 期：2
• 页码：526-539
• 全文大小：565 K

文摘

If the initial and boundary data for a PDE do not obey an infinite set of compatibility conditions, singularities will arise in the solution at the corners of the initial time–space domain. For dissipative equations, such as the 1-D heat equation or 1-D convection–diffusion equations, the impacts of these singularities are short lived. However, they can cause a very severe loss of numerical accuracy if we are interested in transient solutions. The phenomenon has been described earlier from a theoretical standpoint. Here, we illustrate it graphically and present a simple remedy which, with only little extra cost and effort, restores full numerical accuracy.