Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however, a numerical approximation to the solution is often sufficient. 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, including physics, chemistry, biology, and economics. In addition, some numerical methods for partial differential equations convert the partial differential equation into a system of ordinary differential equations, which can then be solved numerically.
Initial value problems An Initial value problem (IVP) is a first-order differential equation plus an initial condition,
where f {\displaystyle f} is a function f : [ t 0 , ∞ ) × R d → R d {\displaystyle f:[t_{0},\infty )\times \mathbb {R} ^{d}\to \mathbb {R} ^{d}} , and y 0 ∈ R d {\displaystyle y_{0}\in \mathbb {R} ^{d}} is a given vector. First-order means that only the first derivative of y {\displaystyle y} appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables. For example, the second-order equation y ″ = − y {\displaystyle y''=-y} can be rewritten as two first-order equations: y ′ = z {\displaystyle y'=z} and z ′ = − y {\displaystyle z'=-y} . In this section, we describe numerical methods for IVPs. Boundary value problems (BVPs) require a different set of tools, discussed in the section below. The Picard–Lindelöf theorem states that the IVP has a unique solution, provided that f {\displaystyle f} is Lipschitz-continuous.
Methods Numerical methods for solving first-order IVPs often fall into one of two large categories: linear multistep methods, which make use of information from previous steps to obtain the next step, or Runge–Kutta methods, which use intermediate stages to calculate the next step. A further division is between explicit and implicit methods. Explicit methods only involve information from the current and previous steps, whereas implicit methods involve the next step, requiring the solution of an equation to solve for the next step. For example, implicit linear multistep methods include Adams-Moulton methods, and backward differentiation methods (BDF), whereas implicit Runge–Kutta methods include diagonally implicit Runge–Kutta (DIRK), singly diagonally implicit Runge–Kutta (SDIRK), and Gauss–Radau (based on Gaussian quadrature) numerical methods. Explicit examples from the linear multistep family include the Adams–Bashforth methods, and any Runge–Kutta method with a lower diagonal Butcher tableau is explicit. A loose rule of thumb dictates that stiff differential equations require the use of implicit schemes, whereas non-stiff problems can be solved more efficiently with explicit schemes. The so-called general linear methods (GLMs) are a generalization of the above two large classes of methods.
Euler method
Euler's method can be thought of as approximating a nearby point on a curve by moving a short distance along a line tangent to the curve. Starting with the differential equation (1), we replace the derivative y ′ {\displaystyle y'} by the finite difference approximation
which when re-arranged yields the following formula
y ( t + h ) ≈ y ( t ) + h y ′ ( t ) {\displaystyle y(t+h)\approx y(t)+hy'(t)}
and using (1) gives:
This formula is usually applied in the following way. We choose a step size h, and we construct the sequence t 0 , t 1 = t 0 + h , t 2 = t 0 + 2 h , … {\displaystyle t_{0},t_{1}=t_{0}+h,t_{2}=t_{0}+2h,\dots } We denote by y n {\displaystyle y_{n}} a numerical estimate of the exact solution y ( t n ) {\displaystyle y(t_{n})} . Motivated by (3), we compute these estimates by the following recursive scheme
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