Summary
A simple systematic method has been developed to investigate the laws of conservation for approximating model equations. The main purpose of this paper is to identify these model equations as approximations of continuous Hamiltonian systems. If this identification is possible, the laws of conservation of the model system can be investigated as for a finite dimensional Hamiltonian system. Obviously, this method can be applied only in the case where the original continuous equations are Hamiltonian. The applicability of the general method has been verified by using three well-known finite-difference schemes as examples. These examples show that this technique is a possible systematic way to construct new conservative finite-difference approximations, as well as to identify the conserved quantities of well-known schemes.
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Szynyogh, I. Finite-dimensional quasi-Hamiltonian structure in simple model equations. Meteorl. Atmos. Phys. 52, 49–57 (1993). https://doi.org/10.1007/BF01025752
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DOI: https://doi.org/10.1007/BF01025752