論具有黏滯性的Burgers方程式數值研究

Abstract

在這篇論文中我們考慮具有黏滯性的Burgers方程式和週期性邊界條件，分析在不同的離散方法下解是否穩定。證明了在半離散的情況下解具有長時間的穩定性；在Engquist-Osher模型下，如果資料具有small data特性，那麼也會有相同的解長時間穩定結果。我們測試了Order accuracy，並且得到EO模型是一階收斂速度。另外測試了長時間的情況下數值情況，得到結果與理論相符。

In this thesis, we consider forced viscous Burgers' equation with periodic boundary conditions. The semi-discrete and fully-discrete schemes are studied. For the semi-discrete scheme, the solution is uniformly bound in the H1 sense provided that time step is small. For the fully-discrete scheme, the solution is uniformly bound in the discrete H1 sense provided that the data and time step are small enough. These proofs are based on the Poincare inequality, Young's inequality and discrete Gronwall lemma. For the numerical tests, we present that the order accuracy for the Engquist-Osher scheme is first and also test the bounds for the velocity and its derivative with different choices of viscosities under L2 norm. The numerical evidences conrm the theoretical results.