Dissipative Methods

For the numerical solution of the problems described in Section 1 a new class of methods has been developed the last three years. These methods are called dissipative methods . 

Definition 5. A method is called dissipative when it is non-symmetric. 

Remark 1. A non-symmetric (i.e. dissipative) multistep method has empty interval of periodicity. 

More details of the above are given in refs. 6 and 16. The development of methods with these properties is an open problem. 

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Unfortunately, the energy dissipation method upon which these calculations are based is only applicable to the evaluation of viscosity in steady deformations. The method does not lend itself to an extension of the disentanglement model to other components of the stress or to other types of deformation history. 

Singer, B. Sh., and E. B. Fainberg, 1995, Generalization of the iterative dissipative method for modeling electromagnetic fields in nonuniform media with displacement currents Journal of Applied Geophysics, 34, 41-46. 

It is also a measure of the moisture stress in plants or soil, measured in megapascals. A more negative value indicates greater moisture stress. Soils with no moisture stress have a water potential of 0 to -1 mPa. Two methods of measuring soil water potential are the heat-dissipation method or the electrical-resistance method. 

Dissipative Methods Developed in the Literature. - For the numerical solution of the second-order periodic problem (81) the following dissipative methods have been developed in the literature during the last two years. 

In Table 6 we present the basic characteristics for the dissipative methods with... 

Numerical Illustrations for Linear Multistep Methods and Dissipative Methods... 

From exponentially fitted and trigonometrically fitted methods the most efficient are the linear symmetric multistep methods. This is because these methods have two additional properties [symmetry (i.e. symplecticness) and non-empty interval of periodicity] to the dissipative methods and one additional property (non-empty interval of periodicity) to the symplectic methods. 

As a conclusion we can say that symmetric (non-dissipative) methods are more efficient because they have non-empty interval of periodicity and because they are symplectie (in the case of linear symmetric multistep methods). We note also that symmetric linear multistep methods are very simple in programming and have very low computational cost (only one function evaluation per step). 

Figure 1. Computed conductivity as a function of applied field. The solid line represents the least squares fit to the high-field method points and the dashed line the mean value of these points. The open circle is the zero field point obtained by the current correlation function method. The crosses are the points obtained by the heat dissipation method.

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