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A local stability analysis is given for both the analytic and numerical solutions of the initial value problem for a system of ordinary differential equations. The standard linear stability analysis is reviewed, then it is shown that, using a proper choice of Liapunov function, a connected region of stable initial values of both the analytic solution and of the one-leg k-step numerical solution can be approximated computationally. Correspondence between the one-leg k-step solution and its associated linear k-step solution is shown, and two examples are given.

A local stability analysis is given for both the analytic and numerical solutions of the initial value problem for a system of ordinary differential equations. The standard linear stability analysis is reviewed, then it is shown that, using a proper choice of Liapunov function, a connected region of stable initial values of both the analytic and one leg k -step numerical solution can be approximated computationally. Correspondence between the one leg k -step solution and its associated linear k -step solution is stated. Constructive proofs are given for the existence of the stability regions for both the analytic and one leg k -step numerical solutions for a class of functions. An example is discussed.

Sufficient conditions for the convergence of methods for the numerical integration of a system of ordinary differential equations, possibly stiff, are presented. The asymptotic error formula for such a solution is developed. The existence of more stringent conditions for convergence of multi-derivative formulas over first derivative formulas is noted. A class of computationally A-stable formulas is presented, with notes on implementing a set of them in a variable stepsize, variable order method. Tests on one such implementation show the capabilities of such methods on stiff problems.

Many important problems in fluid dynamics, among other areas, are modeled by nonlinear parabolic differential systems with initial values given in one 'independent variable' x, and boundary values in the remaining dependent variables. Hyperbolic systems can sometimes be treated as a special case. For example, the inviscid flow case of the Navier-Stokes equations is a hyperbolic system, while the viscous flow case is elliptical. A survey of currently used numerical methods is in Richtmeyer and Morton. In subsonic flow cases, the nonlinear terms are small enough to be ignored, but these terms must be included in supersonic and hypersonic flow. These numerical calculations usually involve a finite difference mesh over the boundary value problem variables, resulting in a space discretization matrix equation which for the nonlinear system varies at each step in x, the independent variable representing time in the dynamic case or one of the space variable for the steady state case. Then this nonlinear system is solved as an initial value problem in x. The initial value problem is usually solved by a one step implicit method for reasons of cost and stability. Some methods based on finite element methods for the boundary value problem can be used, but successful methods are only available for the linear cases, such as subsonic flow problems.

This chapter presents a variable stepsize variable order method that is A-stable for orders up to 7, provided the second and third analytic derivatives can be computed. It presents methods for computing these higher derivatives for linear inhomogeneous systems of ordinary differential equations; it also highlights a similar treatment for equations arising in chemical kinetics. Numerical tests run on sample problems are described, showing the applicability of the treatment. ASTIFF is a FORTRAN program. The numerical method in ASTIFF is well suited to solving the system of equations, y' = Ay + g(t), where A is a matrix, g is a vector valued function of the independent variable, and y is the solution vector. By proper linearization of certain systems of ordinary differential equations, very good approximations to y″ and y″' can be made and thus, ASTIFF can be used to integrate any such system if it is analytically stable, with no restraint on the stepsize other than controlling the truncation error. Time-dependent forcing terms can be easily accounted for by using numerical differentiation of the nonautonomous terms.