Energy inequality

So far we have established an estimate for the rate of convergence in a very simple problem. It is possible to obtain a similar result for this problem by means of several other methods that might be even much more simpler. However, the indisputable merit of the well-developed method of energy inequalities is its universal applicability it can be translated without essential changes to the multidimensional case, the case of variable coefficients, difference schemes for parabolic and hyperbolic equations and other situations. 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

The method of energy inequalities. The well-developed method of energy inequalities from Chapter 2 seems to offer more advantages in investigating the stability of scheme (II) with weights. 

The energy inequality method. An investigation of difference schemes for the string vibration equation may be carried out by means of the energy inequality method (see Section 1). Here we restrict ourselves to stability with respect to the initial data with regard to the problem... 

In this chapter we study the stability with respect to the initial data and the right-hand side of two-layer and three-layer difference schemes that are treated as operator-difference schemes with operators in Hilbert space. Necessary and sufficient stability conditions are discovered and then the corresponding a priori estimates are obtained through such an analysis by means of the energy inequality method. A regularization method for the further development of various difference schemes of a desired quality (in accuracy and economy) in the class of stability schemes is well-established. Numerous concrete schemes for equations of parabolic and hyperbolic types are available as possible applications, bring out the indisputable merit of these methods and unveil their potential. 

The energy identity. We study the problem of stability of scheme (1) by the method of the energy inequalities involving as the necessary manipulations the inner product of both sides of equation (1) with 2rj/ = 2(y — y) ... 

The method of energy inequalities shows in such a setting that the conditions... 

Upon substituting this estimate into (13) we obtain the energy inequality... 

Having at our disposal the energy inequality (57), we can derive the a priori estimates in just the same way as was done for constant operators A and R. For example, for 5 > 0 inequality (57) for problem (la) implies the estimate... 

To take into account the energy inequalities between the participating atomic orbitals the wave functions have extra c terms, as in the example ... 

Figure 8-1 Derivation of Poynting s theorem and the energy inequality.

Taking into account that all integrals on the right-hand side of expression (8.99) are non-negative, we arrive at the important energy inequality,... 

Wc now examine the solution of Maxwell s equation in an unbounded domain with the given distribution of electromagnetic parameters e, //, and electromagnetic field is generated by the sources (extraneous currents j ), concentrated within some local domain Q. Using the energy inequality (8.112), we can prove that there is only one (unique) solution of this problem. 

The proof of this formula is similar to the proof of formula (8.111). Based on formulae (9.6) and (9.7), we can obtain, after some algebraic transformations, the following important energy inequality, derived originally by Singer (1995) and Pankratov et al. (1995) ... 

Similar to frequency domain cases, it can be proved, based on the energy inequality (8.100), that the energy flow Ff" of the residual field is positive ... 

In the winter the situation is quite different. There is a lack of sunlight, and I need more electricity. The batteries are always hungry for a charge in the winter. This energy inequality can be addressed by adding a wind turbine to a PV system. Fortunately, it is often windy at the times when sunlight is lacking, so these power sources complement each other well. 

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