Euler method

The Euler method approximates solutions to ordinary differential equations by advancing in fixed or variable steps along the slope at each current point. It is a first-order method with global error proportional to the step size.

Initial value problem: A differential equation paired with starting values that determine a particular solution. Euler’s step-by-step calculation approximates the solution determined by this equation and starting value.
Explicit midpoint method: A second-order Runge–Kutta method that estimates a step using the slope at its midpoint. It improves on Euler by evaluating an additional slope within each step.
Numerical solution of ordinary differential equations: The computation of approximate solutions to ordinary differential equations using discrete numerical procedures. Euler is a basic procedure for turning an ODE into a sequence of computable values.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work shaped analysis, mechanics, and many areas of mathematics. The method bears Euler’s name and reflects his contributions to differential equations and numerical calculation.
Step size: The time or independent-variable interval between successive numerical approximations. It sets both the size of each Euler update and the scale of its accumulated error.
Heun's method: A second-order explicit Runge–Kutta method that averages the initial and predicted endpoint slopes. Its endpoint-slope correction reduces Euler’s first-order error at modest extra cost.
Exponential growth: Growth in which a quantity’s rate of change is proportional to its current amount. Euler approximates the exponential solution of the equation whose derivative equals the quantity itself.
Numerical analysis: The study of algorithms for obtaining and assessing approximate solutions to mathematical problems. Euler’s method is a foundational example of numerical analysis applied to differential equations.
Local truncation error: The error made by one numerical step when that step begins from the exact solution. The Euler method’s one-step error is generally proportional to the square of the step size.
Runge–Kutta methods: A family of one-step methods that combine multiple derivative evaluations to approximate differential-equation solutions. Euler is the simplest member; higher-order members spend more evaluations to improve accuracy.

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