View source for Difference scheme, viscosity of a

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A concept that characterizes the dissipativeness of difference schemes (see [[#References|[1]]]). The viscosity of a difference scheme shows which supplementary dissipative properties appear in the [[Approximation of a differential equation by difference equations|approximation of a differential equation by difference equations]] (see [[#References|[2]]], [[#References|[3]]]). As well as the term  "viscosity of a difference scheme"  one uses the term  "approximative viscosity"  (see [[#References|[4]]], [[#References|[5]]]). The viscosity of a difference scheme is a [[Dissipative function|dissipative function]] (see [[#References|[6]]]). The structure of the viscosity of a difference scheme is defined by the form of the coefficients with even derivatives of minimal order with respect to the spatial variables in the functions to be calculated, under an expansion of the difference functions in a Taylor series with respect to the grid parameters (see [[#References|[7]]]–[[#References|[9]]]). The coefficients at the third derivatives with respect to the spatial variables are the coefficients (form the matrix) of the dispersion of the difference scheme (see [[#References|[10]]]). The differential representation includes all the terms of the expansion (an infinite number) of the difference operator in a Taylor series with respect to the grid parameters (see [[#References|[9]]], [[#References|[10]]]). A differential approximation includes some of the terms of the expansion; the first differential approximation consists of the initial differential operator and the first non-zero term of the expansion.

Depending on the form of the initial system of differential equations and the type of the basic functions of the expansion, different forms of viscosity and dispersion matrices are realized. In the study of numerical methods of gas dynamics (cf. [[Gas dynamics, numerical methods of|Gas dynamics, numerical methods of]]), there are 6 different forms of viscosity matrices (see [[#References|[10]]]).

The condition of non-negativity of the viscosity matrix of parabolic form of a first differential approximation is considered as a condition of the stability of the difference scheme; in this case one has a [[Well-posed problem|well-posed problem]] (see [[#References|[8]]]). By examining an equation with viscosity of a difference scheme it is possible, using the apparatus of differential approximations, to produce a group classification of difference schemes (see [[#References|[9]]]).

The viscosity of a difference scheme has a unique definition for each definite difference scheme. For an effective control of the viscosity to be possible, it is advisable to examine classes of difference schemes. Thus, by introducing a multi-parameter class of splitting difference schemes (see [[#References|[10]]]), it is possible, by varying the numerical values of the parameters, to change the values of the terms of the viscosity by putting the viscosity in the form of Navier–Stokes, turbulent and other viscosities. Depending on its parameters, the viscosity can be optimized (see [[#References|[11]]]) by the requirement that various conditions of a mathematical, programming and architectural nature be fulfilled. When the conditions of non-negativity and minimality of the viscosity with respect to the parameters of a multi-parameter class of splitting difference schemes are fulfilled, a family of optimal schemes (which are minimally dissipative and stable) can be distinguished; the difference scheme of the [[Large-particle method|large-particle method]] belongs to this family (see [[#References|[12]]]).

In studying the viscosity of a difference scheme it is advisable to reveal the internal structure of the schematic viscosity matrix (see [[#References|[11]]]), for example: to examine the viscosity matrix of a splitting, the non-stationary viscosity matrix, the viscosity matrix of a shift, the architectural viscosity matrix, etc.

In solving a boundary value problem, the concept of the viscosity of a difference scheme and of a differential approximation or of a representation of difference boundary conditions is introduced (see [[#References|[10]]]).

The viscosity of a difference scheme is examined in research into the stability of non-linear difference schemes, both at points within the domain of computation and on the boundaries or in a neighbourhood of them.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.I. Marchuk,   "Methods of numerical mathematics" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Bakhvalov,   "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Samarskii,   Yu.P. Poppov,   "Difference methods for the solution of problems in gas dynamics" , Moscow  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.K. Godunov,   V.S. Ryaben'kii,   "The theory of difference schemes" , North-Holland  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> , ''Theoretical foundations and construction of numerical algorithms of problems of mathematical physics'' , Moscow  (1979)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  O.M. Belotserkovskii,   Yu.M. Davydov,   "Dissipative properties of difference systems" , Moscow  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B.L. Rozhdestvenskii,   N.N. Yanenko,   "Systems of quasilinear equations and their applications to gas dynamics" , Amer. Math. Soc.  (1983)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  N.N. Yanenko,   Yu.I. Shokin,   "On the approximation viscocity of difference schemes"  ''Soviet Math.-Dokl.'' , '''9'''  (1968)  pp. 1153–1155  ''Dokl. Akad. Nauk SSSR'' , '''182''' :  2  (1968)  pp. 280–281</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  Yu.I. Shokin,   "The method of differential approximation" , Springer  (1983)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  Yu.M. Davydov,   "Differential approximations and representations of difference schemes" , Moscow  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  Yu.M. Davydov,   "Structure of approximate viscosity"  ''Sov. Phys.-Dokl.'' , '''24'''  (1979)  pp. 223–226  ''Dokl. Akad. Nauk SSSR'' , '''245''' :  4  (1979)  pp. 812–817</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  O.M. Belotserkovskii,   Yu.M. Davydov,   "The method of large particles in gas dynamics. Numerical experiments" , Moscow  (1982)  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Hedström,   "Models of difference schemes for $u_t+u_x=0$ by partial differential equations"  ''Math. Comp.'' , '''29'''  (1975)  pp. 969–977</TD></TR></table>