Conservative schemes

An example of the scheme which is divergent in the case of discontinuous coefficients. We now consider problem (1) of Section 2.1 with g = 0 and / = 0 incorporated  

As one might expect, the derivative kii ) should be replaced by ku + k u. As a first step towards the construction of a second-order approximation, it will be sensible to carry out the forthcoming substitutions 

Within these notations, a reasonable form of the difference scheme is 

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Reducing scheme (2) to the form (4) from Section 2.1 we find that 

Conditions (5) and (6) from Section 2.1 hold true, since on segments, where the function k(x) is smooth enough, the relations occur  

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The above example shows that in designing difference schemes it is very desirable to reproduce the appropriate conservative law on a grid, The schemes with this property are said to be conservative. In subsequent sections the general method for constructing conservative schemes, which are convergent in the class of discontinuity coefficients, will be appreciated. Before we undertake the complete description of this method, it is worth noting two things. 

From a physical point of view, the finite difference method is mostly based based on the further replacement of a continuous medium by its discrete model. Adopting those ideas, it is natural to require that the principal characteristics of a physical process should be in full force. Such characteristics are certainly conservation laws. Difference schemes, which express various conservation laws on grids, are said to be conservative or divergent. For conservative schemes the relevant conservative laws in the entire grid domain (integral conservative laws) do follow as an algebraic corollary to difference equations. 

We have written the difference equation (14) at a fixed node x = x. With an arbitrarily chosen node it is plain to derive equation (14) at all inner nodes of the grid. Since at all the nodes x, i = 1, 2,.. ., IV — 1, the coefficients a, and are specified by the same formulae (15), scheme (14)-(15) is treated as a homogeneous conservative scheme. Because of this, we may omit the subscript i in formulae (14)-(15) and write down an alternative form of scheme (14) ... 

Observe that the conservation law in the entire grid domain known as the integral conservation law is an algebraic corollary to equation (14) for any conservative scheme of the form (14) with arbitrary ingredients a, d and ip. Indeed, with the notation = —a — y )/h for the... 

Homogeneous conservative schemes. In the preceding section we have... 

At the next stage we consider the homogeneous conservative scheme... 

Conditions (5) of Section 1 relating to the second-order local approximation for the conservative scheme (17) acquire the form... 

In Section 3.2 the integro-interpolational method was aimed at constructing the homogeneous conservative scheme (16) with the coefficients a, d and special form (15), namely with pattern functionals such that... 

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. 

In order to develop a homogeneous conservative scheme on the non-equidistant grid O , we write down the balance equation... 

When only one coefficient k[x) G is discontinuous, while other coefficients q, f are continuous, any conservative scheme (4) generating a second-order approximation is of second-order accuracy on the sequence of non-equidistant grids w (/ ). This fact is an immediate implication of the expansions, — (feu )i j 2 = 0 h ), valid for the... 

Such an approximation is the result of a natural generalization of homogeneous conservative schemes from Chapter 3 for one-dimensional equations to the multidimensional case. These schemes can be obtained by means of the integro-interpolational method without any difficulties. 

Homogeneous difference schemes with weights. In a common setting it seems natural to expect that a difference scheme capable of describing this or that nonstationary process would be suitable for the relevant stationary process, that is, for du/dt = 0 we should have at our disposal a difference scheme from a family of homogeneous conservative schemes, whose use permits us to solve the equation Lu + / = 0. 

One way of covering this for the heat conduction equation is to construct a homogeneous conservative scheme by means of the integro-interpo-lation method. To make our exposition more transparent, we may assume that the coefficient of heat conductivity k = k(x) is independent of t. The general case k = k(x,t) will appear on this basis in Section 8 without any difficulties. 

FuUy conservative schemes. Other ideas are connected with successive use of conservation laws and more detailed balances of the kinetic and internal energy. 

All the schemes with these properties are called fully conservative schemes. As a matter of fact, the requirement of the full conservatism is equivalent to being approximated of both equations (14) and (15) in addition to the usual requirements of approximation ... 

In this connection,there arises a four-parameter family of schemes, from which a fully conservative scheme needs to be selected through the approximations to equations (15) and (10) by appeal to scheme (28)-(29). 

Thus, under such an approach a one-parameter family of fully conservative schemes is given by... 

Further comparison of (40) with (27) shows that a new family of fully conservative schemes is contained in family (25)-(27) of describing conservative schemes with four parameters as a result of employing the integro-interpolation method. 

It is plain to create for this system of equations a fully conservative scheme such as... 

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More precisely, the Fourier coefficients in (4.27) can be replaced by random variables with the following properties k. U = 0 and (U ) = 0 for all k such that k > kc. An energy-conserving scheme would also require that the expected value of the residual kinetic energy be the same for all choices of the random variable. The LES velocity PDF is a conditional PDF that can be defined in die usual manner by starting from die joint PDF for the discrete Fourier coefficients U. ... 

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