Numerical ordinary differential equations - Definition

Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). This field is also known under the name numerical integration, but some people reserve this term for the computation of integrals.

Many differential equations cannot be solved analytically, in which case we have to satisfy ourselves with an approximation to the solution. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution.

Ordinary differential equations occur in many scientific disciplines, for instance in mechanics, chemistry, ecology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.

The problem

We want to approximate the solution of the differential equation

<math>y'(t) = f(t,y(t)), \qquad y(t_0)=y_0, \qquad\qquad (1)<math>

where f is a function that maps [t₀,∞) × R^d to R^d, and the initial condition y₀ ∈ R^d is a given vector.

The above formulation is called an initial value problem (IVP). The Picard-Lindelöf theorem states that there is a unique solution, if f is Lipschitz continuous. In contrast, boundary value problems (BVPs) specify (components of) the solution y at more than one points. Different methods need to be used to solve BVPs, for example the shooting method, multiple shooting or global methods like finite differences or collocation.

Note that we restrict ourselves to first-order differential equations (meaning that only the first derivative of y appears in the equation, and no higher derivatives). However, a higher-order equation can easily be converted to a first-order equation by introducing extra variables. For example, the second-order equation y'' = −y can be rewritten as two first-order equations: y' = z and z' = −y.

Methods

Two elementary methods are discussed to give the reader a feeling for the subject. After that, pointers are provided to other methods (which are generally more accurate and efficient). The methods mentioned here are analysed in the next section.

The Euler method

Starting with the differential equation (1), we replace the derivative y' by the finite difference approximation

<math> y'(t) \approx \frac{y(t+h) - y(t)}{h}, \qquad\qquad (2) <math>

which yields the following formula

<math> y(t+h) \approx y(t) + hf(t,y(t)) \qquad\qquad (3). <math>

This formula is usually applied in the following way. We choose a step size h, and we construct the sequence t₀, t₁ = t₀ + h, t₂ = t₀ + 2h, ... We denote by y_n a numerical estimate of the exact solution y(t_n). Motivated by (3), we compute these estimates by the following recursive scheme

<math> y_{n+1} = y_n + hf(t_n,y_n). <math>

This is the Euler method, named after Leonhard Euler who described this method in 1768.

The backward Euler method

If, instead of (2), we use the approximation

<math> y'(t) \approx \frac{y(t) - y(t-h)}{h}, <math>

we get the backward Euler method:

<math> y_{n+1} = y_n + hf(t_{n+1},y_{n+1}). <math>

The backward Euler method is an implicit method, meaning than we have to solve an equation to find y_n+1. One often uses functional iteration or (some modification of) the Newton-Raphson method to achieve this. Of course, it costs time to solve this equation; this cost must be taken into consideration when one selects the method to use.

Generalisations

The Euler method is often not accurate enough. In more precise terms, it only has order one (the concept of order is explained below). This caused mathematicians to look for higher-order methods.

One possibility is to use not only the previously computed value y_n to determine y_n+1, but to make the solution depend on more past values. This yields a so-called multistep method. Almost all practical multistep methods fall within the family of linear multistep methods, which have the form

<math> \alpha_k y_{n+k} + \alpha_{k-1} y_{n+k-1} + \cdots

+ \alpha_0 y_n<math>

<math> = h \left( \beta_k f(t_{n+k},y_{n+k}) + \beta_{k-1}

f(t_{n+k-1},y_{n+k-1}) + \cdots + \beta_0 f(t_n,y_n) \right). <math>

Another possibility is to use more points in the interval [t_n,t_n+1]. This leads to the family of Runge-Kutta methods, named after Carle Runge and Martin Kutta. One of their fourth-order methods is especially popular.

Both ideas can also be combined. The resulting methods are called general linear methods.

Advanced features

A good implementation of one of these methods for solving an ODE entails more than the time-stepping formula.

It is often inefficient to use the same step size all the time, so variable step-size methods have been developed. Usually, the step size is chosen such that the (local) error per step is below some tolerance level. This means that the methods must also compute an error indicator, an estimate of the local error.

An extension of this idea is to choose dynamically between different methods of different orders (this is called a variable order method). Extrapolation methods are often used to construct various methods of different orders.

Other desirable features include:

• dense output: cheap numerical approximations for the whole integration interval, and not only at the points t₀, t₁, t₂, ...

• event location: finding the times where, say, a particular function vanishes.

• support for parallel computing.

Alternative methods

Many methods do not fall within the framework discussed here. Some classes of alternative methods are:

• multiderivative methods, which use not only the function f but also its derivatives. This class includes Hermite-Obreschkoff methods and Fehlberg methods.

• methods for second order ODEs. We said that all higher-order ODEs can be transformed to first-order ODEs of the form (1). While this is certainly true, it may not be the best way to proceed. In particular, Nyström methods work directly with second-order equations.

• geometric integration methods are especially designed for special classes of ODEs (eg. Hamiltonian equations, reversible equations). They take care that the numerical solution respects the underlying structure or geometry of these classes.

Analysis

Numerical analysis is not only the design of numerical methods, but also their analysis. Three central concepts in this analysis are convergence (whether the method approximates the solution), order (how well it approximates the solution), and stability (whether errors are damped out).

Convergence

A numerical method is said to be convergent if the numerical solution approaches the exact solution as the step size h goes to 0. More precisely, we require that for every ODE (1) with a Lipschitz function f and every t^* > 0,

<math> \lim_{h\to0+} \max_{n=0,1,\dots,\lfloor t^*/h\rfloor}

\| y_{n,h} - y(t_n) \| = 0. <math>

All the methods mentioned above are convergent. In fact, convergence is a condition sine qua non for any numerical scheme.

Order

Suppose the numerical method is

<math> y_{n+k} = \Psi(t_{n+k}; y_n, y_{n+1}, \dots, y_{n+k-1};

h). <math>

The method is said to have order p if

<math> \Psi \left( t_{n+k}; y(t_n), y(t_{n+1}), \dots, y(t_{n+k-1}); h \right)

- y(t_{n+k}) = \mathcal{O}(h^{p+1}). \quad\quad (4) <math>

The quantity on the left-hand side is called the local error of the method. The (forward) Euler method and the backward Euler method introduced above both have order 1. Most methods being used in practise attain higher order.

The local error is the error committed in a single step. A related concept is the global error, the error sustained in all the steps one needs to reach a fixed time t. Explicitly, the global error at time t is y_N - y(t) where N = (t-t₀)/h. The global error of a pth order one-step method (that is, a method of the form (4) with k = 1) is O(h^p); in particular, such a method is convergent. This statement is not necessarily true for multi-step methods.

Stability and stiffness

Loosely speaking, a numerical method is called stable if unwanted components in the numerical solution die out over time. Many different aspects of stability have been discussed in the literature. We will only treat one of them.

A method is A-stable if the numerical results y_n approach zero as n → 0 for all values of the step size h when this method is applied to the equation y' = λy for all λ ∈ C with Re λ < 0. Note that for this equation, the exact solution also goes to zero. The (forward) Euler method is not A-stable, but the backward Euler method is A-stable.

For some differential equations, it does not matter much whether the method is stable. However, for other equations, stable methods perform far better; these equations are said to be stiff (it is hard to formulate a more precise definition). Stiffness is often caused by the presence of different time scales in the underlying problem. Stiff problems are ubiquitous in (chemical) kinetics, control theory, weather prediction, biology, and electronics.

History

Below is a concise timeline of some important developments in this field.

• 1768 - Leonhard Euler publishes his method.
• 1824 - Augustin Louis Cauchy proves convergence of the Euler method. In this proof, Cauchy uses the implicit Euler method.
• 1855 - First mention of the multistep methods of John Couch Adams in a letter written by F. Bashforth.
• 1895 - Carle Runge publishes the first Runge-Kutta method.
• 1905 - Martin Kutta describes the popular fourth-order Runge-Kutta method.
• 1910 - Lewis Fry Richardson announces his extrapolation method.
• 1952 - Charles F. Curtiss and Joseph Oakland Hirschfelder coin the term stiff equations.

References

• Ernst Hairer, Syvert Paul Nørsett and Gerhard Wanner, Solving ordinary differential equations I: Nonstiff problems, second edition, Springer Verlag, Berlin, 1993. ISBN 3-540-56670-8.
• Ernst Hairer and Gerhard Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, second edition, Springer Verlag, Berlin, 1996. ISBN 3-540-60452-9. This two-volume monograph systematically covers all aspects of the field.
• Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996. ISBN 0-521-55376-8 (hardback), ISBN 0-521-55655-4 (paperback). Textbook, targeting advanced undergraduate and postgraduate students in mathematics, which also discusses numerical partial differential equations.
• John Denholm Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, 1991. ISBN 0-471-92990-5. Textbook, slightly more demanding than the book by Iserles.