Higher order gradient methods

There are basically two reasons for this distortion. Standard differencing schemes introduce numerical diffusion to the initial distance function. This problem can be reduced by using higher-order numerical methods. The second reason is that the flow field is very rarely of that character so that the level set function cj) would be kept as a distance function. For example, the maximum and minimum values of the level set function will remain the same throughout the computations. For two merging interfaces (e.g., two bubbles), this will cause a steep gradient and an impenetrable sheet between the two merging interfaces (see Fig. 9). It is therefore necessary to reinitiate the distance function after each time step. 

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

The advantage of the NR method is that the convergence is second-order near a stationary point. If the function only contains tenns up to second-order, the NR step will go to the stationary point in only one iteration. In general the function contains higher-order terms, but the second-order approximation becomes better and better as the stationary point is approached. Sufficiently close to tire stationary point, the gradient is reduced quadratically. This means tlrat if the gradient norm is reduced by a factor of 10 between two iterations, it will go down by a factor of 100 in the next iteration, and a factor of 10 000 in the next ... 

To determine the optimal parameters, traditional methods, such as conjugate gradient and simplex are often not adequate, because they tend to get trapped in local minima. To overcome this difficulty, higher-order methods, such as the genetic algorithm (GA) can be employed [31,32]. The GA is a general purpose functional minimization procedure that requires as input an evaluation, or test function to express how well a particular laser pulse achieves the target. Tests have shown that several thousand evaluations of the test function may be required to determine the parameters of the optimal fields [17]. This presents no difficulty in the simple, pure-state model discussed above. 

Also, the fourth order terms of the gradient expanssion of the kinetic energy have been evaluated [24], leading to more involved expressions. This is one of the problems of the methods based on the gradient expansion the systematic improvement of the results by adding higher orders is not possible because of the asymptotic nature, and... 

Moments and polarizabilities can also be obtained by the fixed-charge method [76]. This technique allows for the single-step incorporation of the nonuniform electric field contributions due to gradients and higher order field derivatives. One or more charges are placed around the molecule in regions where the molecular wavefunctions are negligible. It is important that the basis set used for the field-free molecule be the same as that used in the presence of the field and that the molecule basis be adequate to describe any... 

Nymand et al. ° performed molecular dynamics simulations on liquid water, and they used the electric field effect formalism [Eq. (6)] to explain the gas to liquid shifts of the and O nuclei. For the proton it turned out that the resulting gas to liquid shift of — 3.86 ppm at 300 K compared well with the experimental value of —4.70 ppm, whereas for O the method failed to reproduce the experiment. Even if electric field gradient terms are introduced, requiring additional quadrupolar shielding polarizabilities, no better results could be obtained for the O gas to liquid shifts. Isotropic proton chemical shifts are obviously a special case where many higher order terms cancel, hence it is justified to use the simple electric field equations in these chemical shift calculations. 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

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