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Remark on the paper “On one decomposition scheme for the equations of gas dynamics”
U.S.S.R. Comput.Haths.llath.Phys .,Vo1.26,No.3,pp.98-99,19S6 Printed in Great Britain
0041-5553/86 $~O.GG+G.GC 01987 Pergamon Journals Ltd.
LETTER TO THE EDITOR REMARK ON THE PAPER "ON ONE DECOMPOSITION SCHEME FOR THE EQUATIONS OF GAS DYNAMICS"*
In the author's naper, published in USSR Comp.Maths.vo1.25, No.12, 1985, integral differential equations obtained by integrating the equations of gas dynamics over a fixed volume are considered. For these equations the following decomposition scheme is given: $jmdu+$
fds-0, rt'-% cp'(P,P~,P~), f-(O,P,Pu)*
(W
;i++$
cp(uda)=O,r~'--u,
(lb)
where p is the density u the velocity, e the total energy, and p the pressure of the gas; _ ._. the integrals in the first terms of (1) are over the variable volume, and r is the radius vector of a point that belongs to this variable volume, the integrals in the second terms are over the surface of the variable volume, and da is a surface element that is orientated along the outward normal. A formal transition from Eqs.(l) to equivalent partial differential equations results in a trivial decomposition scheme: (la) becanes the required equations of gas dynamics, (lb) becomes inequalities to zero of the corresponding partial derivatives with respect to time. Later in the paper, using a heuristic prescription that appeals to the procedure for construction on the basis of the difference schemes (l), a decomposition scheme for partial differential equations is derived a+/at+q dig u+vf-0, aaiati (uv)cp-0, that is also considered as an analogue of the decomposition scheme (1). As was shown in a discussion paper at a seminar on methods of solving problems of mathematical physics at the Computing Centre of the USSR Academy of Sciences, the meaning of this analogue gives rise to justified bewilderment. Because of this it is necessary to supplement the paper with the following remark. Suppose there is some difference decomposition scheme constructed using an approximation of Eqs.(l). There is no simple answer to the question of which partial differential equations approximate each stage of the difference scheme. On the same point, the difference scheme i.e. the computational algorithm, that enables us to obtain the value of the required quantities on the higher time layer according to the values of these quantities on the lower layer, in a natural way does not depend on whether the nodes of such a difference grid are related to the intermediate quantities that were obtained in this algorithm. At the same time the partial differential equations that approximate each of the stages do depend on this. In particular, if the intermediate quantities relate to tne nodes of the intermediate difference grid used in the construction of the scheme on the basis of (1). then the stages of the difference scheme will approximate to a trivial decomposition scheme. If the intermediate quantities also relate to the initial grid, then the stages of the difference scheme will approximate to Eq.(2). It is exactly in this sense that there also follows an understanding of the anlogy between (1) and (2). Let us demonstrate one difference scheme family quoted as an example. Suppose there is a difference grid that is developing a domain of flow into kernels. The values of all the unknown functions are determined in the centre of the kernels. We integrate (la) over t from 0 to 7, assuming the volume of integration over a spatial variable for t-0 that coincides with the volume of the difference cell V, and by using the approximate equations
where& is the value of T at the node of the grid on the lower time layer,@ mediate, SO is the surface of the cell, V is the volume of integration over variable for t-7, and U,P are certain expressions depending on the sizes of the approximating functions uand p on the surface of the cell. As a result family of difference schemes of the first stage, that easily gives
*Zh.vychisl.Mat.mat.Fiz.,26,5,793-TN,1986 98
on the interthe spatial 6 and @ and we obtain a
99
(?.a)
Integrating (lb) over t from 0 to 1 and assuming the volume of integration over a spatial variable to be equal to V for r-0 and V for 1-r we similarly obtain the family of difference schemes of the second stage:
where & is the value of cpat the node of the grid on the upper time layer, R,i7,E are certain expressions depending on the values of 6 and @ and the approximating functions p,u and e on the surface of the cell, and U. is the expression approximating u and, possibly, differing from both (I and V. If now we wish to obtain the quantities k,@ and (p with values of some function p at the nodes of some or other grid, then expansion of this function in a Taylor series gives acp & &+;r, WF+;r (4) and the difference Eqs.(3) will approximate the difference scheme (2). If @ relates to the nodes of a deformed grid then the corresponding expansion in a Taylor series takes the form
where r'lt. is the displacement vector of the centre of the cell at time r. As a result the difference Eqs.(3) will approximate the trivial distribution scheme for the partial differential equations. If we formally consider the function cp in expansions (4) and (5) as dependent on time, then both expressions are identical. Therefore precisely that heuristic prescription which is used in the published article results in the distribution scheme (2). A.A. Charakhch'yan Translated by S.R.
0041-5553/86 $~O.GG+G.GC 01987 Pergamon Journals Ltd.
LETTER TO THE EDITOR REMARK ON THE PAPER "ON ONE DECOMPOSITION SCHEME FOR THE EQUATIONS OF GAS DYNAMICS"*
In the author's naper, published in USSR Comp.Maths.vo1.25, No.12, 1985, integral differential equations obtained by integrating the equations of gas dynamics over a fixed volume are considered. For these equations the following decomposition scheme is given: $jmdu+$
fds-0, rt'-% cp'(P,P~,P~), f-(O,P,Pu)*
(W
;i++$
cp(uda)=O,r~'--u,
(lb)
where p is the density u the velocity, e the total energy, and p the pressure of the gas; _ ._. the integrals in the first terms of (1) are over the variable volume, and r is the radius vector of a point that belongs to this variable volume, the integrals in the second terms are over the surface of the variable volume, and da is a surface element that is orientated along the outward normal. A formal transition from Eqs.(l) to equivalent partial differential equations results in a trivial decomposition scheme: (la) becanes the required equations of gas dynamics, (lb) becomes inequalities to zero of the corresponding partial derivatives with respect to time. Later in the paper, using a heuristic prescription that appeals to the procedure for construction on the basis of the difference schemes (l), a decomposition scheme for partial differential equations is derived a+/at+q dig u+vf-0, aaiati (uv)cp-0, that is also considered as an analogue of the decomposition scheme (1). As was shown in a discussion paper at a seminar on methods of solving problems of mathematical physics at the Computing Centre of the USSR Academy of Sciences, the meaning of this analogue gives rise to justified bewilderment. Because of this it is necessary to supplement the paper with the following remark. Suppose there is some difference decomposition scheme constructed using an approximation of Eqs.(l). There is no simple answer to the question of which partial differential equations approximate each stage of the difference scheme. On the same point, the difference scheme i.e. the computational algorithm, that enables us to obtain the value of the required quantities on the higher time layer according to the values of these quantities on the lower layer, in a natural way does not depend on whether the nodes of such a difference grid are related to the intermediate quantities that were obtained in this algorithm. At the same time the partial differential equations that approximate each of the stages do depend on this. In particular, if the intermediate quantities relate to tne nodes of the intermediate difference grid used in the construction of the scheme on the basis of (1). then the stages of the difference scheme will approximate to a trivial decomposition scheme. If the intermediate quantities also relate to the initial grid, then the stages of the difference scheme will approximate to Eq.(2). It is exactly in this sense that there also follows an understanding of the anlogy between (1) and (2). Let us demonstrate one difference scheme family quoted as an example. Suppose there is a difference grid that is developing a domain of flow into kernels. The values of all the unknown functions are determined in the centre of the kernels. We integrate (la) over t from 0 to 7, assuming the volume of integration over a spatial variable for t-0 that coincides with the volume of the difference cell V, and by using the approximate equations
where& is the value of T at the node of the grid on the lower time layer,@ mediate, SO is the surface of the cell, V is the volume of integration over variable for t-7, and U,P are certain expressions depending on the sizes of the approximating functions uand p on the surface of the cell. As a result family of difference schemes of the first stage, that easily gives
*Zh.vychisl.Mat.mat.Fiz.,26,5,793-TN,1986 98
on the interthe spatial 6 and @ and we obtain a
99
(?.a)
Integrating (lb) over t from 0 to 1 and assuming the volume of integration over a spatial variable to be equal to V for r-0 and V for 1-r we similarly obtain the family of difference schemes of the second stage:
where & is the value of cpat the node of the grid on the upper time layer, R,i7,E are certain expressions depending on the values of 6 and @ and the approximating functions p,u and e on the surface of the cell, and U. is the expression approximating u and, possibly, differing from both (I and V. If now we wish to obtain the quantities k,@ and (p with values of some function p at the nodes of some or other grid, then expansion of this function in a Taylor series gives acp & &+;r, WF+;r (4) and the difference Eqs.(3) will approximate the difference scheme (2). If @ relates to the nodes of a deformed grid then the corresponding expansion in a Taylor series takes the form
where r'lt. is the displacement vector of the centre of the cell at time r. As a result the difference Eqs.(3) will approximate the trivial distribution scheme for the partial differential equations. If we formally consider the function cp in expansions (4) and (5) as dependent on time, then both expressions are identical. Therefore precisely that heuristic prescription which is used in the published article results in the distribution scheme (2). A.A. Charakhch'yan Translated by S.R.