APPLICATION OF THE VARIATIONAL PRINCIPLES OF MECHANICS TO CONSTRUCT TIME-DISCRETE DIFFERENCE GAS-DYNAMIC MODELS.8. IMPLICIT FINITE-DIFFERENCE SCHEMES WITH PHASE VOLUME AND CANONICITY PRESERVED

Yu. A. Bondarenko
VANT. Ser.: Mat. Mod. Fiz. Proc 2011. Вып.2. С. 3-17.

It has been proved for the finite-difference schemes of Lagrangian gas dynamics built with the variational method using the time- and space-discrete definition of the principle of least action (Hamilton-Ostrogradskii principle) that they preserve phase volume and canonicity (Hamiltonicity). The paper also proves that implicit finite-difference schemes with constant weight = 1/2 (in equations for coordinates and velocity) do not preserve phase volume and, far less, canonicity for an arbitrary, variable time step with any way of choosing the hidden generalized coordinates and hidden generalized momentum (such difference schemes could not be built with the stepwise variational method).

Keywords: Lagrangian gas dynamics, principle of least action, variational difference schemes, schemes with weights, variable time step, phase volume, canonicity, hidden variables.

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