Numerical integration singularities

The formulation above allows a more general equation of state for the combustion products (Kuhl 1983). The method described breaks down for low piston velocities, where the leading shock Mach number approaches unity. In such cases, the numerical integration marches into the point (F = 0, Z = 1), which is a singularity. 

Unlike the previous case the integrand in eq. 4.166 does not have singularities, and this expression is convenient for calculations. Some results of numerical integration are presented in Table 4.3. As these data show, corrections are usually small and do not exceed 10% even when the radius of the receiver ring is equal to half of the borehole radius. With an increase of the radius the geometric factor, G, decreases, and it is specially noticeable for relatively small probes. It is obvious that with an increase of the probe the influence of the ring radius on the geometric factor GJ decreases. 

Nevertheless, when singularities occur, it is sometimes observed that calculated primary current distributions, obtained with the combination of analytical and numerical integration, are worse than the less accurate classical integration method where only points belonging... 

Let, as in Section 2, (y z) denote the solution of the ODE system (2.4) and (yo,zo) the solution of the DAE system (2.5). Then the QSSA error of interest after one integration step is a = 11(2/) )( ) (2/Oj o)(t), where r is the timestep chosen by the applied numerical integrator. In the special situation, we can apply standard results from singular perturbation theory, in particular a quite well-known result of Vasilyeva [21] - see, for instance, the textbook [19]. If we assume the right-hand side F to be at least twice differentiable, the following asymptotic expansion is known to hold ... 

At the bifurcation points of the system, Eq. (7.142), the Jacobian matrix becomes singular. Hence, for numerical integration of Eq. (7.147), a parameterization procedure along the curve x(F) can be used in the corresponding space of dimension dim(x+1). This integration requires the use of special methods, in particular, methods based on the calculation of the Jacobian... 

Integrals with singularities over the range of integration present special problems for numerical integration techniques. Some examples of such integrals are ... 

According to the Cauchy theorem, the integral is zero if Em Eo, because the singularity Em is not inside F, and it is zero when Em = Eo because the singu "Tity of the denominator is compensated by the numerator. Therefore, R o and K are zero. 

It is noticed that the Green s function A (x) has one singularity when Xy = 0, but it can be eliminated by an integration over the element around the point Xy=0. A computation experiment shows that Eq (30) may result in a significant numerical error when coarser grids are employed. 

In Eq (25), the integration kernel function K x)= rr x is smooth everywhere except the singularity point (x = 0). In a numerical analysis, the integration has to be evaluated in discrete form over a grid with the mesh size h... 

The numerical evaluation of definite integrals can be carried out in several ways. However, in all cases it must be assumed that the function, as represented by a table of numerical values, or perhaps a known function, is well behaved. While this criterion is not specific, it suggests that the functions haying pathological problems, e.g. singularities, discontinuities,..may not survive under the treatment in question. 

The cumulative functions of the diffusive modes can here also be constructed by using Eq. (60) with trajectories integrated with a numerical algorithm based on the rescaling of time at the singular collisions. The initial position is taken on a small circle around a scattering center at an angle 0 with respect to the... 

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