Explicit scheme

The difference problem (34 ) illustrates the implementation of the so-called explicit scheme in which the values of the solution on the upper layer j/2+1 are expressed through the values on the current layer by the explicit formulae... 

On the grid u>f arising from Section 2.1 it is simple to follow the explicit scheme of accuracy 0(/i + r) ... 

The explicit scheme is stable only under condition (25) relating the grid steps h and r (a conditionally stable scheme). 

We give a brief survey afforded by the above results scheme (II) converges uniformly with the same rate as in the grid L2(w )-norm (see (35)) if and only if condition (39) holds. The stability condition (39) in the space C for the explicit scheme with cr = 0, namely r < coincides with the... 

A case in point is that for the explicit scheme with a = 0 the uniform convergence does not follow from (46) under the constraint t < h. But the a priori estimate emerged in Section 5.7, namely... 

Whence it follows immediately that the explicit scheme converges uniformly... 

The third kind boundary conditions. The first kind boundary conditions we have considered so far are satisfied on a grid exactly. In Chapter 2 we have suggested one effective method, by means of which it is possible to approximate the third kind boundary condition for the forward difference scheme (a = 1) and the explicit scheme (cr = 0) and generate an approximation of 0 t -b h ). Here we will handle scheme (II) with weights, where cr is kept fixed. In preparation for this, the third kind boundary condition... 

This provides support for the view that the solution is completely distorted. From such reasoning it seems clear that asymptotic stability of a given scheme is intimately connected with its accuracy. When asymptotic stability is disturbed, accuracy losses may occur for large values of time. On the other hand, the forward difference scheme with cr = 1 is asymptotically stable for any r and its accuracy becomes worse with increasing tj, because its order in t is equal to 1. In practical implementations the further retention of a prescribed accuracy is possible to the same value for which the explicit scheme is applicable. Hence, it is not expedient to use the forward difference scheme for solving problem (1) on the large time intervals. 

This provides support for the view that the explicit scheme with cr = 0 is stable ... 

Explicit schemes for the Cauchy problem. The first-order equation... 

Explicit schemes of a higher-order approximation. Of major importance is the explicit scheme of accuracy 0 h r) having the form... 

The two-layer scheme (la) with constant operators A and B can be reduced to the explicit scheme... 

In order to prove this theorem, we beforehand reduce the implicit scheme (la) to the explicit scheme (32) with the operator C —... 

Under this condition estimate (15) holds true for a solution of problem (47). In particular, for an explicit scheme (for cr = 0) the condition cr > (Tq implies r < 2/ A, that is, an explicit scheme is stable in the space Ha for r < 2/11 A. A scheme with r > is unconditionally stable, that is,... 

Three-layer schemes with non-self-adjoint operators. The three-layer explicit scheme with a self-adjoint operator A... 

The explicit scheme (82) for the Schrodinger equation takes the form... 

For the explicit scheme with tr = 0 this implies that... 

Having stipulated this condition, the explicit scheme y f = y ) for the string vibration equation is stable for r/h < l/- /(l + s) (see Chapter... 

The next step is to reduce the resulting scheme to more convenient forms in trying to avoid cumbersome calculations. The case tr = 0, relating to the explicit scheme, is simple to follow ... 

This condition is only sufficient for the indicated property. A necessary and sufficient condition for the stability of the explicit scheme with respect to the initial data in the space Ha is... 

For this, it seems unreasonable to employ explicit schemes with fastly varying ingredients k(u), c(u) and f(u). The power functions of temperature reflect in full measure the diiflculties involved in such a case. For any implicit scheme one possible stability condition... 

From here it seems clear that the admissible step in the explicit scheme is yet to be refined along with increasing the maximum value of the coefficient of heat conductivity. As a matter of fact, the last requirement is unreal for the problems with fastly and widely varying coefficients. Just for this reason explicit schemes are of little use not only for multidimensional problems, but also for one-dimensional ones (p = 1). On the other hand, the explicit schemes ofl er real advantages that the value y = on every new layer + t is found by the explicit formulas (3) with a finite... 

Some consensus of opinion is desirable in this matter, since a smaller number of operations is performed in the explicit scheme, but it is stable only for sufficiently small values of r. In turn, the implicit scheme being absolutely stable requires much more arithmetic operations. 

What schemes are preferable for later use Is it possible to bring together the best qualities of both schemes in line with established priorities In other words, the best scheme would be absolutely stable as the implicit schemes and schould require in passing from one layer to another exactly Q arithmetic operations. As in the case of the explicit schemes, Q would be proportional to the total number of the grid nodes so that Q = 0 l/hf). 

Here the passage from the jth layer to the j + l)th layer is carried out in the following two steps the first one involves the explicit scheme and the second one - the implicit scheme as suggested above. 

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