Overview
Adams-Bashforth methods extrapolate from past derivative evaluations, reducing new function evaluations after startup.
ExtendedSource checked
Adams-Bashforth Method
Solves ODEs with an explicit multistep formula using previous derivative values.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Advanced
- Family
- ODE Solver
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 4
- Code example
- Ready
- Visualization
- Ready
Core Concepts
multistep methodderivative historyexplicit updateode integration
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(state)
Average
O(steps * state)
Worst
O(steps * state)
Space
O(history)
When to Use
- Use when solving equations needs multistep method.
- Use when the input matches the assumptions behind oDE Solver.
When Not to Use
- Avoid when the problem is ill-conditioned and no scaling or stability checks are available.
- Avoid when a simpler algorithm gives the required result with less setup.
What It Does
- Solves ODEs with an explicit multistep formula using previous derivative values
- Uses multistep method and derivative history to guide the work.
- Produces approximate numeric result.
Use Cases
- Solving equations
- Approximating integrals
- Working with matrices and differential equations
Inputs
- Numeric function, matrix, or sampled data
- Initial estimate or discretization
- Tolerance
Outputs
- Approximate numeric result
- Error estimate or residual
Key Formulas (1)
Two-step Adams-Bashforth
ynext = yn + dt · (3·fn − f_{n−1}) / 2
y_next = y_n + dt * (3*f_n - f_{n-1}) / 2Uses the current and previous derivative to advance the solution.
Pseudocode
- 1.Prepare the inputs and state needed for Adams-Bashforth Method.
- 2.Apply the multistep method step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return approximate numeric result.
Canonical Example
- Input
- Adams-Bashforth Method input with a small representative case
- Output
- Approximate numeric result
- Explanation
- Adams-Bashforth Method applies multistep method to produce approximate numeric result.
Implementation Notes
- Check convergence with residuals or estimated error, not iteration count alone.
- Keep the multistep method step separate enough to test independently.
Common Pitfalls
- Ignoring conditioning and residuals can produce plausible-looking but wrong numeric answers.
- Forgetting to check edge cases can make Adams-Bashforth Method look correct on only ideal inputs.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow the numeric approximation improving against the target function, matrix, or state.
Step 1 of 4: Prepare the inputs and state needed for Adams-Bashforth Method.
numerical-methods-mechanismNotes
Adams-Bashforth Method should be used with the assumptions and tradeoffs of oDE Solver in mind.
Sources (1)
What each reference supports for this entry.
Solving Ordinary Differential Equations I
BookReference text covering explicit multistep methods including Adams-Bashforth formulas.
Explore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageAlternative toBackward Differentiation Formula
Both are multistep ODE methods with different stability profiles.
Alternative toRunge-Kutta Method
Adams-Bashforth uses previous derivative values instead of only within-step stages.
Related Algorithms
Tags
Numerical MethodsOde SolverExplicit MethodNumerical Integration