Methods and Applications of Analysis

Volume 10 (2003)

Number 2

Discrete Approximations with Additional Conserved Quantities: Deterministic and Statistical Behavior

Pages: 151 – 190



Rafail V. Abramov

Andrew J. Majda


Discrete numerical approximations with additional conserved quantities are developed 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 symbiotic interaction of mathematical theory and scientific computing. The new contributions include an explicit concrete discussion of the sine-bracket spectral truncation with many conserved quantities 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.

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