Convergence order finite difference approximations

It is apparent that as the momentum p increases, the finite difference spectrum deviates more and more from the correct value. It is usually assumed that acceptable accuracy with the FD method is obtained when at least 10 points are used per wave period. This means also using 10 points per unit volume in phase space. The finite difference algorithms are based on a local polynomial approximation of the wave function and therefore the convergence of the method follows a power law of the form (Aq)n, where n is the order of the finite difference approximation. This semilocal description leads to a poor spectral representation of the kinetic energy operator, which will be true as well, for other banded representations of the kinetic energy operator such as the... 

Note the rapid convergence using different initial guesses. The MAT-LAB M-file tangent. m (Figure 2.21) has been created in order to implement the Newton method. It uses a finite difference approximation to the first derivative... 

A differential equation that has data given at more than one value of the independent variable is a boundary-value problem (BVP). Consequently, the differential equation must be of at least second order. The solution methods for BVPs are different compared to the methods used for initial-value problems (IVPs). An overview of a few of these methods will be presented in Sections 6.2.1. 2.3. The shooting method is the first method presented. It actually allows initial-value methods to be used, in that it transforms a BVP to an IVP, and finds the solution for the IVP. The lack of boundary conditions at the beginning of the interval requires several IVPs to be solved before the solution converges with the BVP solution. Another method presented later on is the finite difference method, which solves the BVP by converting the differential equation and the boundary conditions to a system of linear or non-hnear equations. Finally, the collocation and finite element methods, which solve the BVP by approximating the solution in terms of basis functions, are presented. 

For our purposes, CC theory and its finite order MBPT approximations offer a convenient, compact description of the correlation problem and give rapid convergence to the basis set (i.e. full Cl) limit with different categories of correlation operators (see Fig. 15.1). The coupled-cluster wave function is... 

A property of the finite volume method is that numerous schemes and procedures can be designed in order to solve the two-fluid model equations. In addition, the coupling terms can be approximated and manipulated in different ways. Besides, it is very difficult to predict the convergence and stability properties of novel solution methods. These aspects collectively increase the possibility of devising a numerical procedure which, when put to the test, does not converge. 

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