Second order difference method

In the present section a direct method for solving the boundary-value problems associated with second-order difference equations will be the subject of special investigations. 

With these, for determination of yt = j/l + 1 we obtain a second-order difference equation supplied by the boundary conditions (71)—(72) that can be solved by the standard elimination method. 

The truncation error for Eq. (10.40) is the same as those stated for the momentum equation for p = 0, A, 1. The fully implicit scheme can be increased to a formal second-order accuracy by representing the streamwise derivatives with three-level (i-l,i,i+l) second-order differences. For any implicit method, the finite difference momentum and energy equations are algebraically nonlinear in the unknowns because of the quantities unknown at the i+1 level in the coefficients. Linearizing procedures can and have been used, but are beyond the scope of this book. 

The spurious or satellite term in the solution is introduced by using a second-order difference equation to approximate a first-order differential equation. An extra condition is needed to fix the solution of the second-order equation, and this condition must be that the coefficient of the spurious part of the solution is zero. In the general case of a nonlinear difference equation, no method is available for meeting this condition exactly. 

The use of difference methods offers a means whereby a detailed picture of ionic hydration can be obtained 22). For neutron diffraction, the first-order isotopic difference method (see Section III,A) provides information on ionic hydration in terms of a linear combination of weighted ion-water and ion-ion pair distribution functions. Since the ion-water terms dominate this combination, the first-order difference method offers a direct way of establishing the structure of the aquaion. In cases for which counterion effects are known to occur, as, for example, in aqueous solutions of Cu + or Zn +, it is necessary to proceed to a second difference to obtain, for example, gMX and thereby possess a detailed knowledge of both the aquaion-water and the aquaion-coun-terion structure. 

Fig. 6 Average binding energy (Eb, eV) and second-order difference in total energy (A , eV) of the clusters using the composite G3B3 method. The b and values of neutral boron cluster B (n = 2-20) obtained from Refs. [20] and [26]...

A different approach is to represent the wavepacket by one or more Gaussian functions. When using a local harmonic approximation to the trae PES, that is, expanding the PES to second-order around the center of the function, the parameters for the Gaussians are found to evolve using classical equations of motion [22-26], Detailed reviews of Gaussian wavepacket methods are found in [27-29]. 

A different long-time-step method was previously proposed by Garci a-Archilla, Sanz-Serna, and Skeel [8]. Their mollified impulse method, which is based on the concept of operator splitting and also reduces to the Verlet scheme for A = 0 and admits second-order error estimates independently of the frequencies of A, reads as follows when applied to (1) ... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Method of Variation of Parameters This technique is applicable to general linear difference equations. It is illustrated for the second-order system -2 + yx i + yx = ( )- Assume that the homogeneous solution has been found by some technique and write yY = -I- Assume that a particular solution yl = andD ... 

The truncation error in the first two expressions is proportional to Ax, and the methods are said to be first-order. The truncation error in the third expression is proportional to Ax, and the method is said to be second-order. Usually the last equation is used to insure the best accuracy. The finite difference representation of the second derivative is ... 

B. van Leer, Towards the Ultimate Conservative Difference Scheme. V.A. Second-Order Sequel to Godunov s Method, J. Comput. Phys. 32 (1979). 

---