Math 2.0

Numerical analysis Concept

Euler method

Builds on Initial value problemLipschitz conditionDiscrete Grönwall inequality

Summary

The Euler method is a numerical method to approximate the solution of an initial value problem y′=f(t,y)y' = f(t, y), y(t0)=y0y(t_0) = y_0. With a step size h>0h > 0, it sets tk+1=tk+ht_{k+1} = t_k + h and yk+1=yk+h f(tk,yk)y_{k+1} = y_k + h\, f(t_k, y_k) for k=0,1,2,…k = 0, 1, 2, \ldots. Then yky_k approximates y(tk)y(t_k). It follows the tangent line of the solution for one step at a time.

Remarks

The error of one step, the local truncation error, is h22y′′(τ)\frac{h^2}{2} y''(\tau) for some τ\tau between tkt_k and tk+1t_{k+1}, when yy is twice continuously differentiable. Let ff be LL-Lipschitz in yy with L>0L > 0, and let ∣y′′∣≤M|y''| \le M on the interval. Then the global error satisfies ∣y(tk)−yk∣≤hM2L(eL(tk−t0)−1)|y(t_k) - y_k| \le \frac{h M}{2 L} \left( e^{L (t_k - t_0)} - 1 \right). The proof uses a discrete Grönwall inequality. So the method is of first order: halving the step size roughly halves the error. It is the simplest Runge–Kutta method. Leonhard Euler proposed it in his book Institutionum calculi integralis.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionDiscrete Grönwall inequality, statement, no evidence levelDiscrete Grönwall ine…statementInitial value problem, definition, no evidence levelInitial value problemdefinitionEuler method, concept, no evidence levelEuler methodconcept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Grönwall's inequality statement
  7. Solution of an ordinary differential equation definition
  8. Discrete Grönwall inequality statement
  9. Initial value problem definition
  10. Euler method concept

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Euler_method, Derivation
  2. Web https://en.wikipedia.org/wiki/Euler_method, Local truncation error
  3. Web https://en.wikipedia.org/wiki/Euler_method, Global truncation error
About this page
Tag0JARTXX4XP
URIm2://0JARTXX4XP
Aliasesforward-euler-method, explicit-euler-method
Kindconcept
WritingW1 stated
MSC65L05, 65L20
UsesInitial value problem, Lipschitz condition, Discrete Grönwall inequality
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge defines in: at least one definition (0 of 1)
  • edge instance_of in: at least two examples (0 of 2)
  • edge not_instance_of in: at least one non-example (0 of 1)
  • edge specializes both: the more general notions and the special cases (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by