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1.2.5 Moments and conservation laws

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Differential calculus is at the heart of science and engineering because it describes interactions locally, relating infinitesimal changes at the microscopic scale to the macroscopic behavior a system. At the macroscopic scale, global quantities can be found that remain constant in spite of the microscopic changes: the total density, the momentum and the energy inside a closed box that is isolated from the outside world do not change. Conservation laws can in general be constructed by taking moments in phase space $ x$ , where the moment of order $ K$ a function $ f(x)$ is defined by the integral

$\displaystyle \mathcal{M}_K = \int_\Omega dV x^K f$ (1.2.5#eq.1)

Usually, $ \mathcal{M}_0$ refers to the total density, $ \mathcal{M}_1$ the total momentum and $ \mathcal{M}_2$ the total energy in the system. Conservation laws provide useful self-consistency checks when PDEs are solved in an approximate manner with the computer: deviations from the initial value are a clear signal for the loss of accuracy in a numerical solution that evolves in time.



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