Convergence and Error Estimates for the Lagrangian-Based Conservative Spectral Method for Boltzmann Equations
Convergence and Error Estimates for the Lagrangian-Based Conservative Spectral Method for Boltzmann Equations
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DOI：
10.1137/18m1173332
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发表时间：
2016-11
期刊：
影响因子：
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通讯作者：
R. Alonso;I. Gamba;S. H. Tharkabhushanam
中科院分区：
文献类型：
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作者：
R. Alonso;I. Gamba;S. H. Tharkabhushanam
本文对Gamba和Tharkabhushanam在[J]中引入的弹性和非弹性空间齐次Boltzmann方程的半离散保守谱方法进行了误差估计。第一版。理论物理。， 228 (2009), pp. 2012—2036]。此外，我们还研究了平衡麦克斯韦分布的这种半离散解的长时间收敛性，该分布保存了与初始数据相关的质量、动量和能量。数值方法基于碰撞算子的傅里叶变换和拉格朗日优化校正，该校正强制执行碰撞不变量，即弹性情况下质量、动量和能量守恒，而非弹性情况下只有质量和动量守恒。本文对所提出的数值方法的收敛性进行了详细的半离散分析，其中包括该方案的$L^{1}-L^{2}$理论。这个分析允许我们提出，另外，在Sobolev空间收敛和收敛到平衡的数值ap。
We develop error estimates for the semidiscrete conservative spectral method for the approximation of the elastic and inelastic space homogeneous Boltzmann equation introduced by Gamba and Tharkabhushanam in [J. Comput. Phys., 228 (2009), pp. 2012--2036]. In addition we study the long time convergence of such semidiscrete solution to the equilibrium Maxwellian distribution that conserves the mass, momentum, and energy associated with the initial data. The numerical method is based on the Fourier transform of the collisional operator and a Lagrangian optimization correction that enforces the collision invariants, namely, conservation of mass, momentum, and energy in the elastic case, and just mass and momentum in the inelastic one. We present a detailed semidiscrete analysis on convergence of the proposed numerical method which includes the $L^{1}-L^{2}$ theory for the scheme. This analysis allows us to present, additionally, convergence in Sobolev spaces and convergence to equilibrium for the numerical ap...