Euler’s method for numerical integration
E54272
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
Aliases (3)
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
explicit method
→
first-order method → initial value problem solver → numerical method → one-step method → |
| accuracyDependsOn |
step size h
→
|
| alsoKnownAs |
Euler’s method
→
forward Euler method → |
| appliedIn |
engineering computations
→
physics simulations → population dynamics models → |
| approximates |
solution curve of dy/dt = f(t,y)
→
|
| assumes |
given initial condition y(t0) = y0
→
|
| convergenceOrder |
1
→
|
| doesNotRequire |
solving algebraic equations at each step
→
|
| errorType |
accumulated round-off error
→
truncation error → |
| geometricInterpretation |
uses tangent line approximation at each step
→
|
| globalErrorOrder |
O(h)
→
|
| hasUpdateFormula |
y_{n+1} = y_n + h f(t_n, y_n)
→
|
| input |
differential equation dy/dt = f(t,y)
→
initial time t0 → initial value y0 → step size h → |
| isExplicit |
true
→
|
| isGeneralizedBy |
Heun’s method
→
classical fourth-order Runge–Kutta method → higher-order Runge–Kutta methods → improved Euler method → midpoint method → |
| isNotWellSuitedFor |
stiff differential equations
→
|
| isRelatedTo |
backward Euler method
→
predictor–corrector methods → semi-implicit Euler method → |
| isSpecialCaseOf |
Runge–Kutta methods
→
linear multistep methods with one step → |
| isSuitableFor |
non-stiff differential equations
→
|
| localTruncationErrorOrder |
O(h^2)
→
|
| namedAfter |
Leonhard Euler
→
|
| output |
discrete approximation (t_n, y_n) to solution
→
|
| requires |
evaluation of derivative function f at current point
→
|
| smallerStepSizeEffect |
increases accuracy
→
increases computational cost → |
| stabilityType |
conditionally stable
→
|
| timeSteppingDirection |
forward in time
→
|
| typicalUse |
simple simulations where high accuracy is not critical
→
teaching basic concepts of numerical ODE solving → |
| usedFor |
approximating solutions of ordinary differential equations
→
numerical integration of initial value problems → |
| uses |
step size h
→
|
Referenced by (4)
| Subject (surface form when different) | Predicate |
|---|---|
|
Euler’s method for numerical integration
("Euler’s method")
→
Euler’s method for numerical integration ("forward Euler method") → |
alsoKnownAs |
|
Euler–Maruyama method
("Euler method")
→
|
basedOn |
|
Leonhard Euler
→
|
notableWork |