J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 85
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In the problems weconsider later, however, such boundary conditions will occur in theform of conditions prescribed at a certain fixed point of the river interms of the time: for example, the height, or stage, of the river mightbe given at a certain station as a function of the time. In other words,WATER WAVES474conditions would be prescribed not only along the #-axis of our x, t0) for a certainplane, but also along the -axis (in general only for tfixed value of x. The method of finite differences used in Chapter 10.2>to discuss the initial value problem, with the general result given above,in an obvious way to deal with cases in which bound-can be modifiedary conditions are also imposed.
In doing so, it would also becomeclear just what kind of boundary conditions could and should be imposed. For example, in the great majority of rivers in fact, for allwhich the flow is subcritical, i.e. such that v is everywhere less thaninthewave speed \/^jyitis possible to prescribe only one conditionthewhichJ-axis,alongmight be either the velocity v or the depth y,in contrast with the necessity to impose two conditions along the ay-This fact would become obvious on setting up the finite difference scheme, and examples of it will be seen later on.Finally, it should be stated that the role of the characteristics, andaxis.also themethod offinite differences applied tothem could be usedwith reference to the general case of the characteristic equations asembodied in the equations (11.4.7) and (11.4.8) in essentially thesame way as was sketched out above for the system comprised of(11.4.8) and (11.4.9) which referred to a special case.
In particular,the role of the characteristics as curves along which small disturbancespropagate, and their role in determining the domain of dependence,range of influence, etc. remain the same.11.5.Numerical methods for calculating solutions of theequations for flow in open channelsdifferentialIt has already been stated that while the formulation of our problems by the method of characteristics is most valuable for studyingmany questions concerned with general properties of the solutionsof the differential equations, it is in most cases not the best formula-tion to use for the purpose of calculating the solutions numerically.That is not to say that the device of replacing derivatives by differ-ence quotients should be given up, but rather that this device shouldbe used in a different manner.
The basic idea is to operate with finitex /-plane, in conwhich the net ofbeinthethesolutionistoatwhichxpointsapproximated isJ-planedetermined only gradually in the course of the computation. In thelatter procedure it is thus necessary to calculate not only the valuesdifferencesby using a fixed rectangular netmethod outlined in Chaptertrast with the9in the10.2, in9MATHEMATICAL HYDRAULICS475unknown functions v and c, but also the values of the coordinatesof the net points themselves, whereas a proceduremaking use of afixed net would require the calculation of v and cand it wouldof thex,tonly,alsohave the advantage of furnishing these values at a convenientsetof points.However, the question of the convergence of the approximatesolution to the exact solution when the mesh width of a rectangularnet is made to approach zero is more delicate than it is when the method of characteristics is used.
For example, it is not correct, in general,to choose a net in which the ratio of theand the mesh width Ax along themesh width At along the /-axisis kept constant independentof the solution: such a procedure would not in general yield approximations converging to the solution of the differential equation pro-r-axisblem. The reason for this can be understood with reference to one ofthe basic facts about the solution of the differential equations whichwas brought outfact in questionpreceding section. The basicthe existence of what was called there the domainin the discussion of theisof dependence of the solution. For example, suppose the solution wereto be approximated at the points of the net of Fig. 11. 5.
la by advanc-ing from one row parallel to the tT-axis to the next row a distance Atfrom it. In addition, suppose this were to be done by determining theapproximate values of v andc atany point suchP (cf. Fig.as11.5.1b)P\C2I2xb11.5.1.Approximation by using a rectangular netby using the values of these quantities at the nearest three points0, 1, 2 in the next line below, replacing derivatives in the two differential equations by difference quotients, and then solving the resultingIt seems reasonalgebraic equations for the two unknowns v p and c pable to suppose that such a scheme would be appropriate only if Pwere in the triangular region bounded by the characteristics drawnfrom points and 2 to form the region within which the solution is de.WATER WAVES476termined solely by the data given on the segment 02: otherwise itseems clear that the initial values at additional points on the #-axisought to be utilized since our basic theory tells us that the initial dataat some of them would indeed influence the solution at point P.
On theother hand, the characteristic curves themselves depend upon thevalues of the unknown functions v and c their slopes, in fact, arevc and thus the interval At must begiven (cf. (11.4.8)) by dxjdtchosen small enough in relation to a fixed choice of the interval Axso that the points such as P will fall within the appropriate domains ofdeterminacy relative to the points used in calculating the solution atP. In other words, the theory of characteristics, even if it is not useddirectly, comes into play in deciding the relative values of At and Axwhich will insure convergence (for rigorous treatments of thesequestions see the papers by Courant, Isaacson, and Rees [C.ll], andby Keller and Lax[K.4]).We shall introduce two different schemes employing the method ofa fixed rectangular net of the x, J-plane. The firstmakes use of the differential equations in the form given by(11.4.7), and we no longer suppose that the function E is restricted infinite differences inof theseany way.
(It might be noted that the slopes of the characteristics asgiven by (11.4.8) are determined by the quantities v dr c, no matterhow the function E is defined, and in fact also for the most generalcase of a river having a variable cross section A etc., and hence we arein a position to determine appropriate lengths for the ^-intervals, in9accord with the above discussion, in the most general case. This isalso a good reason for working with the quantity c in place of y.)At the same time, the calculation is based on assuming that the ap-Fig. 11.5.2.proximate values ofL,M, R(cf.cArectangular netand v have been calculated at the net pointsand that the differential equations are to beFig. 11.5.2)MATHEMATICAL HYDRAULICS477used to advance the approximate solution to the point P. Thediffer-ential equations to be solved are thus+ v)c x + c } + {(c + v)v x +v }+E = 0,- 2{(- c + v)c x + c + {(- c + v)v x + v } + E = 0,(11.5.2)and the characteristic directions are determined by dx/dt = vThe characteristic with slope v + c we call the forward characteristic,(11.5.1)2{(cttt}tc.and that with slopevthe backward characteristic.cWeshall re-place the derivatives in the equations by difference quotients whichapproximate the values of the derivatives at the point M.
In order toadvance the values of v and c from the points L,R to the point Pandisitnaturalto(11.5.2)by using (11.5.1)replace the time deriva-Mtives v tandby the followingctKxni(11.5.3)y,,=9difference quotientsvM-__,Vpci=cpCMzrHowever, in order to insure the convergence ofand At ->the approximations to the exact solution when Ax ->foraofthisandReesfact) it is[C.ll]proof(see Courant, Isaacson,vandcdifferencex byquotientsnecessary to replace the derivatives xwhich are defined differently for (11.5.1) than for (11.5.2), as follows:in both equations.vx=V(11.5.4)^L^Ax9Cx= ^_I_^(11.5.5).=*-*=!*Ax,c.=Axin (11.5.1),C-*^-**AxThe reason for this procedure is, at bottom, thatin (11.5.2).(11.5.1)isan equationassociated with the forward characteristic, while (11.5.2) is associatedwith the backward characteristic.
The coefficients of the derivativesand the functionarc, of course, to be evaluated at the point M. TheEdifference equations replacing (11.5.1)n.,e,and(11.5.2) are thus givenby,,.+<+ "J^!L )+(,+) !*-pAxAtu = 0,c)}(,= o.WATEE WAVES478We observe that the two unknowns, v p and c poccur linearly in thesefoundarehenceeasilyby solving the equations.
Thetheyequations;result,isVP(11.5.8)^V M +Cp-c M +|(11.5.9)In accordance with the remarksmade above, we must also require thatAx be taken small enough so that P lies within thetriangle formed by drawing lines from L and R in the directions of theforward and backward characteristics respectively, i.e. lines with theC R at jK: a condition that is well-deslopes V L + CL at L and V Rtermined since the values of v and c are presumably known at L and R.One can now see in general terms how the initial value problem= can be solved approximately: One starts with a netstarting at tthe#-axis with spacing Ax. Since c and v arc known at all ofalongthese points, the values of c and v can be advanced through use of(11.5.8) and (11.5.9) to a parallel row of points on a line distant Atalong the 2-axis from the a?-axis. However, the mesh width At mustbe chosen small enough so that the convergence condition is satisfiedat all net points where new values of v and c are computed.We can now see also how to take care of boundary conditions, i.e.of conditions imposed at a fixed point (say at the origin, x = 0) asthe ratio of At togiven functions of the time.
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