DISCRETE APPROXIMATIONS WITH ADDITIONAL CONSERVED QUANTITIES: DETERMINISTIC AND STATISTICAL BEHAVIOR ∗
DISCRETE APPROXIMATIONS WITH ADDITIONAL CONSERVED QUANTITIES: DETERMINISTIC AND STATISTICAL BEHAVIOR ∗
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具有额外保守量的离散近似：确定性和统计行为*
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
A. Majda
中科院分区：
文献类型：
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作者：
R. Abramov;A. Majda
本文提出了具有附加守恒量的正压地球物理流动的离散数值近似，推广了二维不可压缩流体方程，并对Burgers-Hopf方程进行了适当的离散化。通过数学理论和科学计算的共生相互作用，在各种不同的背景下研究这些近似的数学、数值和统计性质。新的贡献包括对二维不可压缩流具有许多守恒量的正弦括号谱截断的明确具体讨论，与标准谱截断的理论和数值比较，以及随着傅里叶模态数量增加而收敛到适当弱解的严格证明。建立了Burgers-Hopf方程的系统离散化，它保留了线性动量和非线性能量；关于这些模型的统计行为的仔细数值实验表明，它们是遍历的和强混合的，但在傅里叶空间中不具有能量均分。此外，在单个网格点上的值的概率分布可以是高度非高斯分布，在该分布中有两个强孤立峰。这与保存二次能量的差分格式的统计行为的早期结果形成对比。通过对Galerkin-truncated Burgers-Hopf模型的一个新的案例研究，介绍了统计相关守恒量的问题。
Discrete numerical approximations with additional conserved quantities are devel- oped here both for barotropic geophysical flows generalizing the 2D incompressible fluid equations and suitable discretizations of the Burgers-Hopf equation. Mathematical, numerical, and statistical properties of these approximations are studied below in various different contexts through the sym- biotic interaction of mathematical theory and scientific computing. The new contributions include an explicit concrete discussion of the sine-bracket spectral truncation with many conserved quanti- ties for 2D incompressible flow, a theoretical and numerical comparison with the standard spectral truncation, and a rigorous proof of convergence to suitable weak solutions in the limit as the number of Fourier modes increases. Systematic discretizations of the Burgers-Hopf equation are developed, which conserve linear momentum and a non-linear energy; careful numerical experiments regarding the statistical behavior of these models indicate that they are ergodic and strongly mixing but do not have equipartition of energy in Fourier space. Furthermore, the probability distribution of the values at a single grid point can be highly non-Gaussian with two strong isolated peaks in this distribution. This contrasts with earlier results for statistical behavior of difference schemes which conserve a quadratic energy. The issues of statistically relevant conserved quantities are introduced through a new case study for the Galerkin-truncated Burgers-Hopf model.