Energy Conserving Methods

The purpose of this paper is twofold (i) We summarize possible difficulties with the midpoint method (other than resonance instability, which has been treated extensively elsewhere) by looking at a simple (molecular) model problem, (ii) We investigate the suitability of some energy conserving methods. 

Time-reversible energy conserving methods can be obtained by appropriate modifications to the (time-reversible) midpoint method. Two such modifications are (i) scaling of the force field by a scalar such that total energy... 

The energy conserving method (6) is a close variation of the method... 

Note that, in loeal eoordinates. Step 2 is equivalent to integrating the equations (13). Thus, Step 2 can either be performed in loeal or in eartesian coordinates. We consider two different implicit methods for this purpose, namely, the midpoint method and the energy conserving method (6) which, in this example, coineides with the method (7) (because the V term appearing in (6) and (7) for q = qi — q2 is quadratie here). These methods are applied to the formulation in cartesian and in local coordinates and the properties of the resulting propagation maps are discussed next. 

We apply the semi-implicit algorithm to handle the weak potentials Vi, and the energy conserving method (16) for the stiff forces. The maximal error in the total energy, i.e. 

A difficulty with the energy conserving method (6), in general, is the solution of the corresponding nonlinear equations [6]. Here, however, using the initial iterate (q + A p , p ) for (q +i, p +i), even for large values of a we did not observe any difficulties with the convergence of Newton s method. 

Unfortunately, discretization methods with large step sizes applied to such problems tend to miss this additional force term [3]. Furthermore, even if the implicit midpoint method is applied to a formulation in local coordinates, similar problems occur [3]. Since the midpoint scheme and its variants (6) and (7) are basically identical in local coordinates, the same problem can be expected for the energy conserving method (6). To demonstrate this, let us consider the following modified model problem [13] ... 

Greene, R. (Ed.), Process Energy Conservation Methods and Technology, reprints from Chemical Engineering, McGraw-Hill Publications, New York (1982). 

Another reason why energy conserving methods are rarely used in molecular simulation is that they may lead to behavior that is not consistent with the properties of our original model. If we think of the procedure that we use to define the step as defining a map of phase space, the map does not necessarily have qualitative properties that mimic those of the original system. Note that, as we have mentioned... 

Warm-edge technology (energy conservation) Methods of separating two glass sheets (window pane) around the edge using a medium that has low thermal-transport properties. 

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