Primary family

In the theory of difference schemes with a primary family of schemes the coefficients of a homogeneous difference scheme are expressed through the coefficients of the initial differential equation by means of the so-called pattern functionals the arbitrariness in the choice of these functionals is limited by the requirements of approximation, solvability, etc. There are various ways of taking care of these restrictions. The availability of a primary family of homogeneous difference schemes is ensured by a family of admissible pattern functionals known in advance. 

The principal question in the theory of homogeneous difference schemes is connected with further design of admissible schemes within a primary family for solving a class of typical problems as wide as possible and choosing the most efficient ones (in accuracy, volume of computations, etc.). 

A primary family of conservative schemes. We spoke above about the family of the homogeneous conservative schemes (17), whose description is connected with some class of pattern functionals j4[ (s)] and C[/(s)]. For... 

In what follows we deal everywhere with the primary family of homogeneous conservative schemes (16), (17) and (16 ), (17) as well as with linear nonnegative pattern functionals j4[ (s)] and i [/(s)] still subject to conditions (20) and (21) of second-order approximation. 

As can readily be observed, these conditions remain valid for any scheme from the primary family. 

A primary family of schemes. We will discover stability in a certain... 

Conditions (l)-(3) in combination with the solvability requirement single out a family of admissible schemes known as a primary family from the set of all possible schemes (1). Observe that condition 1) can be weakened in a number of different ways. Sometimes we will deal with operators A and B, which are dependent on t, that is, withA = A t) and B = B t). 

Suppose now the operator A > 0 not to be self-adjoint. Then scheme (18) does not belong to the primary family. However, it can be replaced by an equivalent scheme from the primary family. Since A > 0, there exists an inverse operator A > 0, whose use with regard to equation (18) permits us to confine ourselves to... 

Theorem 4 If condition (14) is satisfied, then scheme (1) from the primary family of schemes is stable with respect to the right-hand side and for a solution of problem (1) the a priori estimate holds ... 

A primary family of describing schemes is specified by the following restrictions ... 

These restrictions permit us to specify and extract a primary family of schemes,... 

Stability or instability of a scheme from the primary family depends only on selection rules for the operator R. From the point of view of stability theory the arbitrariness in the choice of the operator R is restricted by the following requirements ... 

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

This condition can easily be verified in the case of discrete schemes for equations of mathematical physics. It allows one to extract from the primary family of schemes the set of stable schemes, within which one should look for schemes with a prescribed accuracy, volume of computations, and other desirable properties and parameters. 

A primary family of schemes. We will discover stability in a certain primary family of difference schemes. Before going further, we regard operators A and B to be bounded linear operators defined on the entire space Hh, X>(A) = V(B) = Hh- In what follows the difference problem (1) is presupposed to be solvable for any input data y0 and that is, there... 

As before we assume the existence of an inverse operator 5-1(f), which assures us of solvability of problem (1) for any input data y0 and [Pg.421]

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