Concept Architecture
Concept
Theoretically, the Euler Method is a first-order numerical integration technique for approximating the solution of an ordinary differential equation (ODE) subject to an initial value problem. It is founded on the principle that the local behaviour of a differentiable function can be approximated by its tangent, allowing the solution to be advanced incrementally over small step sizes. The method provides the simplest numerical approach for solving differential equations when analytical solutions are unavailable or impractical.
Mathematically, the Euler Method approximates the solution by evaluating the derivative at the beginning of each step and projecting the solution forward using a linear approximation. The numerical solution converges to the true solution as the step size decreases, although the method exhibits first-order global accuracy and accumulated truncation error.
In practice, the Euler Method is used to approximate dynamic systems represented by differential equations, including disease progression models, pharmacokinetic models, population dynamics and other continuous-time health economic simulations. Although largely superseded by higher-order methods for production analyses, it remains an important introductory numerical method and is occasionally used where computational simplicity is prioritised.
Purpose
Used to obtain approximate numerical solutions to ordinary differential equations when closed-form analytical solutions are unavailable, providing a simple iterative framework for modelling continuously changing systems.
Mathematical Formulae
Primary Formula
y??? = y? + h ? f(t?, y?)
Supporting Formulae
Initial condition:
y(t?) = y?
Local truncation error:
LTE = O(h�)
Global error:
Error = O(h)
Related Mathematical Methods
- Initial Value Problem
- Numerical Integration
- Runge-Kutta Methods
- Predictor-Corrector Methods
- Adaptive Step-Size Methods
Example
A disease progression model is defined by:
dy/dt = ?0.20y
Initial prevalence:
y? = 1.00
Time step:
h = 0.5 years
The first Euler approximation is:
y? = 1.00 + 0.5 ? (?0.20 ? 1.00)
y? = 0.90
The estimated disease prevalence after the first six-month interval is therefore 0.90.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
+ | =B2+C2*D2 | Performs one Euler update using the current value, step size and derivative. |
POWER | =POWER(1-rate,time) | Compares Euler approximations with analytical solutions where available. |
ABS | =ABS(B3-E3) | Calculates approximation error against a reference solution. |
SEQUENCE | =SEQUENCE(21,1,0,0.5) | Generates equally spaced time intervals for numerical integration. |
VBA (Optional)
Automate iterative Euler calculations across multiple time steps to simulate continuous disease or economic system dynamics.
Sources
- Butcher JC. Numerical Methods for Ordinary Differential Equations.
- Burden RL, Faires JD, Burden AM. Numerical Analysis.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes.
Related Concepts (3)
Library
Publications
1
Numerical Analysis of Parameterized Nonlinear Equations — Werner C. Rheinboldt, 1st Edition ed., 1986 (John Wiley & Sons)
A specialist numerical-analysis reference on iterative solution, continuation and computation for parameterized nonlinear systems, linked here to the ontology's numerical-method and optimization branch.
BookView source →
Frequently Asked Questions (6)
What is the Euler method?
A numerical method for approximating the solution of an initial value problem by advancing in small steps using the local gradient.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
How does the Euler method approximate a solution?
The Euler method approximates the solution of an initial value problem by advancing in small steps, using the local gradient at each step to estimate the next value. Starting from the known initial condition, it moves forward a short distance along the slope given by the differential equation, then repeats. This step-by-step advance guided by the local gradient is how the Euler method builds an approximate solution As the simplest step-by-step solver, the Euler method is easy to understand and apply but takes only one gradient per step, so it trades accuracy for simplicity against more elaborate schemes.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
What does the Euler method use at each step?
At each step, the Euler method uses the local gradient, the slope given by the differential equation at the current point, to decide the direction and size of the advance. It takes a small step along that slope to reach the next estimated value. Relying on the local gradient to advance in small steps is the essence of how the Euler method proceeds As the simplest step-by-step solver, the Euler method is easy to understand and apply but takes only one gradient per step, so it trades accuracy for simplicity against more elaborate schemes.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
What problem does the Euler method solve?
The Euler method solves an initial value problem, approximating its solution by advancing in small steps from the given starting conditions using the local gradient. Because the initial value problem supplies a starting point and a rule for the slope, the Euler method can step forward from that point. Solving an initial value problem in this stepwise way is the method's purpose As the simplest step-by-step solver, the Euler method is easy to understand and apply but takes only one gradient per step, so it trades accuracy for simplicity against more elaborate schemes.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
Why does the Euler method advance in small steps?
The Euler method advances in small steps because it uses the local gradient, which is only accurate near the current point, so a short step keeps the estimate close to the true curve. Larger steps would let the approximation drift further from the solution between evaluations of the gradient. Taking small steps is therefore how the Euler method limits the error introduced at each advance As the simplest step-by-step solver, the Euler method is easy to understand and apply but takes only one gradient per step, so it trades accuracy for simplicity against more elaborate schemes.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
How does the Euler method relate to the Runge-Kutta method?
The Euler method approximates the solution of an initial value problem by advancing in small steps using the local gradient, while the Runge-Kutta method is a family of methods that combine multiple estimates within each step to improve accuracy. The Euler method uses a single gradient per step, whereas Runge-Kutta methods use several. The two are connected as step-by-step solvers of ordinary differential equations that differ in accuracy per step As the simplest step-by-step solver, the Euler method is easy to understand and apply but takes only one gradient per step, so it trades accuracy for simplicity against more elaborate schemes.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 2 Apr 2026
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/euler-method
- Term code
- CS-DE-NOM-001
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