Central difference operator

Equivalently, in terms of the central difference operator 8X, we can write... 

Explicit different schemes show poor stability properties (Mitchell, 1969). In terms of the central difference operator, it may be shown that an accurate implicit equation is... 

The solutions to the three separate equations 3.2, 3.3, and 3.4 can be computed by solving only tridiagonal equations since the left sides of the equations involve only three-point central difference operators 8x,82,8y,82,8z,82 and 5. Although the right side of equation 3.3 involves the product of these operators, it does not complicate the solution process since they are applied to the known solution values from the previous time step. 

For the discretization of spatial derivatives, the particular method uses a family of central difference operators, while a parametric expression with an extra degree of freedom for the temporal derivatives is employed [57], In two dimensions, these approximants have the general forms... 

Assuming a central difference operator for stress [Pg.520]

D = differential operator I = integral operator E = shift operator A = forward difference operator V = backward difference operator 6 = central difference operator p = averager operator. 

With these introductory concepts in mind, let us proceed to develop the backward, forward, and central difference operators and the relationships between these and the differential operators. 

The relationships between central difference operators and differential operators can now be developed, Eq. (3.73), representing the first averaged central difference, is combined with Eqs. (3.17) and (3.22) to yield... 

The higher-order averaged odd central difference operators are obtained by taking products of Eqs. (3.78) and (3.81). The higher-order even central differences are formulated by taking powers of Eq. (3.81). The third and fourth central operators, thus obtained, are listed below ... 

In order to develop the inverse relationships, i.e., equations for the differential operators in terms of the central difference operators, we must first derive an algebraic relationship between p and 8. To do this, we start with Eqs. (3.72) and (3.80). Squaring both sides of Eq. (3.72), we obtain... 

The complete set of relationships between central difference operators and differential operators is summarized in Table 3.3. These relationships will be used in Chap. 4 to develop a set of formulas expressing the derivatives in terms of central finite differences. These formulas will have higher accuracy than those developed using backward and forward finite differences. 

Express the differential and averaged central difference operators in terms of Iheir respective definitions ... 

Stability theory is the central part of the theory of difference schemes. Recent years have seen a great number of papers dedicated to investigating stability of such schemes. Many works are based on applications of spectral methods and include ineffective results given certain restrictions on the structure of difference operators. For schemes with non-self-adjoint operators the spectral theory guides only the choice of necessary stability conditions, but sufficient conditions and a priori estimates are of no less importance. An energy approach connected with the above definitions of the scheme permits one to carry out an exhaustive stability analysis for operators in a prescribed Hilbert space Hh-... 

It was seen that if yi is not chosen to make C practically zero, the solution of Eq. (5-16) oscillates widely. The same requirements apply to the subsequent steps thus, if a significant error is introduced at any step by rounding, that error will be magnified as the solution proceeds. For this reason, a high accuracy in the individual numerical operations is required when a difference equation that suffers from this sort of instability is used. In using this central-difference approximation with nonlinear equations, however, the problem of getting started with a proper value of yi is usually more serious than the problem of controlling roundoff errors. 

Differencing is an operation on a sequence which gives another sequence. We can therefore apply it again to that result sequence. This gives the sequence of second differences. 5(5V) is naturally denoted by 52V, and 52 is called the second difference operator. If we use the central difference convention... 

This is the sequence [..., 0, —1,1,0,...]. Convolving another sequence with it gives the sequence of first differences. Again, a yfz can position the sequence in the right place for central differences if aligning sequences is really important. Again, we can give this operator a name ... 

This shows that the central-difference approximation can be thought of as an operator that filters out scales smaller than the mesh size. Furthermore, the approximation yields the derivative of an averaged value of v x). 

This robust higher order finite-difference method, originally presented in [10,13,25], develops a seven-point spatial operator along with an explicit six-stage time-advancing technique of the Runge-Kutta form. For the former operator, two central-difference suboperators are required a) an antisymmetric... 

Reproducibility Reproducibility expresses the precision under conditions where the results are obtained by the same analytical procedure on identical samples in different laboratories, and possibly with different operators and different equipment. This is rarely used when the tendency is to use central laboratories. 

The derivative (D) being approximated by the finite-difference operator (FD) to within a truncation error (TE) (or, discretization error). The foregoing mathematical consideration provides an estimate of the accuracy of the discretization of the difference operators. It shows that TE is of the order of (Ax)2 for the central difference, but only O(Ax) for the forward and backward difference operators of first order. Equations (4.41) and (4.42) involve 2 or 3 nodes around node i at x , leading to 2- and 3-point difference operators. Considering additional Taylor series expansions extending to nodes i + 2 and i - 2 etc., located at x + 2Ax and x. — 2Ax, etc., respectively, one may derive 4- and 5-point difference formulas with associated truncation errors. Results summarized in Table 4,8 show that a TE of O(Ax)4 can be achieved in this manner. The penalty for this increased accuracy is the increased complexity of the coefficient matrix of the resulting system of equations. 

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