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 in an ordinary differential equation, which must then be solved.
Table of contents |
2 Methods
2.1 The Euler method
3 Analysis2.2 The backward Euler method 2.3 Generalisations 2.4 Advanced features 2.5 Alternative methods 4 History 5 References |
We want to approximate the solution of the differential equation
where f is a function that maps [t_{0},∞) × R^{d} to R^{d}, and the initial condition y_{0} ∈ 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.
Starting with the differential equation (1), we replace the derivative y' by the finite difference approximation
If, instead of (2), we use the approximation
we get the backward Euler method:Both ideas can also be combined. The resulting methods are called general linear methods.
Other desirable features include:
Many methods do not fall within the framework discussed here. Some classes of alternative methods are:
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).
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,
Suppose the numerical method is
The method is said to have order p if
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_{(t-t0)/h} - y(t). 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.
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.
Below is a concise timeline of some important developments in this field.