NAG C Library Chapter Introduction

d02 — Ordinary Differential Equations

These conditions would be sufficient to define a solution in the range 0≤x≤10, but the problem could not be solved by direct integration (see Section 2.2). More general boundary conditions are permitted in the boundary-value case.

A special class of initial value problems are those for which the solutions contain rapidly decaying transient terms. Such problems are called stiff; an alternative way of describing them is to say that certain eigenvalues of the Jacobian matrix ∂fi ∂yj   have large negative real parts when compared to others. These problems require special methods for efficient numerical solution; the methods designed for non-stiff problems when applied to stiff problems tend to be very slow, because they need small step lengths to avoid numerical instability. A full discussion is given in Hall and Watt (1976) and a discussion of the methods for stiff problems is given in Berzins et al. (1988).

In general, a system of nonlinear differential equations with boundary conditions at two or more points cannot be guaranteed to have a solution. The solution, if it exists, has to be determined iteratively. A comprehensive treatment of the numerical solution of boundary-value problems can be found in Ascher et al. (1988) and Keller (1992). The methods for this chapter are discussed in Ascher et al. (1979), Ascher and Bader (1987) and Gladwell (1987).

If a boundary-value problem seems insoluble by the above methods and a good estimate for the solution of the problem is known at all points of the range then a finite-difference method may be used. Finite-difference equations are set up on a mesh of points and estimated values for the solution at the grid points are chosen. Using these estimated values as starting values a Newton iteration is used to solve the finite-difference equations. The accuracy of the solution is then improved by deferred corrections or the addition of points to the mesh or a combination of both. The method does not suffer from the difficulties associated with the shooting method but good initial estimates of the solution may be required in some cases and the method is unlikely to be successful when the solution varies very rapidly over short ranges. A discussion is given in Chapters 9 and 11 of Gladwell and Sayers (1980) and Chapter 4 of Gladwell (1979a).

In general, for non-stiff first-order systems, Runge–Kutta (RK) functions should be used. For the usual requirement of integrating across a range the appropriate functions are nag_ode_ivp_rk_setup (d02pvc) and nag_ode_ivp_rk_range (d02pcc); nag_ode_ivp_rk_setup (d02pvc) is a setup function for nag_ode_ivp_rk_range (d02pcc). For more complex tasks there are a further four related functions: nag_ode_ivp_rk_onestep (d02pdc), nag_ode_ivp_rk_reset_tend (d02pwc), nag_ode_ivp_rk_interp (d02pxc) and nag_ode_ivp_rk_errass (d02pzc). When a system is to be integrated over a long range or with relatively high accuracy requirements the variable-order, variable-step Adams codes may be more efficient. The appropriate function in this case is nag_ode_ivp_adams_gen (d02cjc). For more complex tasks using an Adams code there are a further four related functions: nag_ode_ivp_adams_roots (d02qfc), nag_ode_ivp_adams_setup (d02qwc), nag_ode_ivp_adams_free (d02qyc) and nag_ode_ivp_adams_interp (d02qzc).

For stiff systems, that is those which usually contain rapidly decaying transient components, the Backward Differentiation Formula (BDF) variable-order, variable-step codes should be used. The appropriate function in this case is nag_ode_ivp_bdf_gen (d02ejc).

If you are not sure how to classify a problem, you are advised to perform some preliminary calculations with nag_ode_ivp_rk_range (d02pcc), which can indicate whether the system is stiff. We also advise performing some trial calculations with nag_ode_ivp_rk_range (d02pcc) (RK), nag_ode_ivp_adams_gen (d02cjc) (Adams) and nag_ode_ivp_bdf_gen (d02ejc) (BDF) so as to determine which type of function is best applied to the problem. The conclusions should be based on the computer time used and the number of evaluations of the derivative function fi. See Gladwell (1979b) for more details.

The basic RK function is nag_ode_ivp_rk_onestep (d02pdc) which takes one integration step at a time. An alternative is nag_ode_ivp_rk_range (d02pcc), which provides output at user-specified points. The initialization of either nag_ode_ivp_rk_range (d02pcc) or nag_ode_ivp_rk_onestep (d02pdc) and the setting of optional inputs, including choice of method, is made by a call to the setup function nag_ode_ivp_rk_setup (d02pvc). Optional output information about error assessment, can be obtained by calls to the diagnostic function nag_ode_ivp_rk_errass (d02pzc). nag_ode_ivp_rk_interp (d02pxc) may be used to interpolate on information produced by nag_ode_ivp_rk_onestep (d02pdc) to give solution and derivative values between the integration points. nag_ode_ivp_rk_reset_tend (d02pwc) may be used to reset the end of the integration range whilst integrating using nag_ode_ivp_rk_onestep (d02pdc).

The general Adams variable-order variable-step function is nag_ode_ivp_adams_roots (d02qfc), which provides a choice of automatic error control and the option of a sophisticated root-finding technique. The initialization of nag_ode_ivp_adams_roots (d02qfc) and the setting of optional inputs is made by a call to the setup function nag_ode_ivp_adams_setup (d02qwc). nag_ode_ivp_adams_interp (d02qzc) may be used to interpolate on information produced by nag_ode_ivp_adams_roots (d02qfc) to give solution and derivative values between the integration points.

There is a simple driving function nag_ode_ivp_adams_gen (d02cjc), which integrates a system over a range and, optionally, computes intermediate output and/or determines the position where a specified function of the solution is zero.

There is a simple driving function nag_ode_ivp_bdf_gen (d02ejc), which integrates a system over a range and, optionally, computes intermediate output and/or determines the position where a specified function of the solution is zero. It has a specification similar to the Adams function nag_ode_ivp_adams_gen (d02cjc) except that to solve the equations arising in the BDF method an approximation to the Jacobian ∂fi ∂yj   is required. This approximation can be calculated internally but you may supply an analytic expression. In most cases supplying a correct analytic expression will reduce the amount of computer time used.

For simple boundary-value problems with assigned boundary-values you may prefer to use a code based on the finite difference method for which there is a function with simple calling sequence (nag_ode_bvp_fd_nonlin_fixedbc (d02gac)).

For difficult boundary-value problems, where you need to exercise some control over the calculation, and where the collocation method proves unsuccessful, you may wish to try the alternative method of finite-differences (nag_ode_bvp_fd_nonlin_gen (d02rac)).

Note that it is not possible to make a fully automatic boundary-value function, and you should be prepared to experiment with different starting values or a different function if the problem is at all difficult.

nag_ode_bvp_fd_nonlin_fixedbc (d02gac) may be used for simple boundary-value problems with assigned boundary-values.

You may find that convergence is difficult to achieve using nag_ode_bvp_fd_nonlin_fixedbc (d02gac) since only specifying the unknown boundary-values and the position of the finite-difference mesh is permitted. In such cases you may use nag_ode_bvp_fd_nonlin_gen (d02rac), which permits specification of an initial estimate for the solution at all mesh points and allows the calculation to be influenced in other ways too. nag_ode_bvp_fd_nonlin_gen (d02rac) is designed to solve a general nonlinear two-point boundary-value problem with nonlinear boundary conditions.

A function, nag_ode_bvp_fd_lin_gen (d02gbc), is also supplied specifically for the general linear two-point boundary-value problem written in a standard ‘textbook’ form.

You are advised to use interpolation functions from Chapter e01 to obtain solution values at points not on the final mesh.