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Integration of Ordinary Differential Equations part 5

ym=vector(1,nvar); yn=vector(1,nvar); h=htot/nstep; Stepsize this trip. for (i=1;i | 724 Chapter 16. Integration of Ordinary Differential Equations ym vector 1 nvar yn vector 1 nvar h htot nstep Stepsize this trip. for i 1 i nvar i ym i y i yn i y i h dydx i First step. x xs h derivs x yn yout Will use yout for temporary storage of deriva- h2 2.0 h tives. for n 2 n nstep n General step. for i 1 i nvar i swap ym i h2 yout i ym i yn i yn i swap x h derivs x yn yout for i 1 i nvar i Last step. yout i 0.5 ym i yn i h yout i free_vector yn 1 nvar free_vector ym 1 nvar CITED REFERENCES AND FURTHER READING Gear C.W. 1971 Numerical Initial Value Problems in Ordinary Differential Equations Englewood Cliffs NJ Prentice-Hall 6.1.4. Stoer J. and Bulirsch R. 1980 Introduction to NumericalAnalysis New York Springer-Verlag 7.2.12. 16.4 Richardson Extrapolation and the Bulirsch-Stoer Method The techniques described in this section are not for differential equations containing nonsmooth functions. For example you might have a differential equation whose right-hand side involves a function that is evaluated by table look-up and interpolation. If so go back to Runge-Kutta with adaptive stepsize choice That method does an excellent job of feeling its way through rocky or discontinuous terrain. It is also an excellent choice for quick-and-dirty low-accuracy solution of a set of equations. A second warning is that the techniques in this section are not particularly good for differential equations that have singular points inside the interval of integration. A regular solution must tiptoe very carefully across such points. Runge-Kutta with adaptive stepsize can sometimes effect this more generally there are special techniques available for such problems beyond our scope here. Apart from those two caveats we believe that the Bulirsch-Stoer method discussed in this section is the best known way to obtain high-accuracy solutions to ordinary differential equations with minimal computational effort. A possible exception infrequently encountered in practice is discussed in .