Overview
Backward Differentiation Formula methods use previous solution values and an implicit current derivative, often requiring a nonlinear solve per step.
SpecializedSource checked
Backward Differentiation Formula
Solves stiff ODEs with implicit multistep formulas based on backward differences.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Advanced
- Family
- ODE Solver
- Tier
- Specialized
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 3
- Code example
- Ready
- Visualization
- Ready
Core Concepts
implicit multistepstiff equationbackward differencenonlinear solve
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(solve cost)
Average
O(steps * solve cost)
Worst
O(steps * solve cost)
Space
O(history)
When to Use
- Use when solving equations needs implicit multistep.
- 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 stiff ODEs with implicit multistep formulas based on backward differences
- Uses implicit multistep and stiff equation 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)
First-order BDF
ynext = yn + dt · f(tnext, ynext)
y_next = y_n + dt * f(t_next, y_next)Uses an implicit Euler-style equation for the next state.
Pseudocode
- 1.Prepare the inputs and state needed for Backward Differentiation Formula.
- 2.Apply the implicit multistep 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
- Backward Differentiation Formula input with a small representative case
- Output
- Approximate numeric result
- Explanation
- Backward Differentiation Formula applies implicit multistep to produce approximate numeric result.
Implementation Notes
- Check convergence with residuals or estimated error, not iteration count alone.
- Keep the implicit multistep 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 Backward Differentiation Formula 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 Backward Differentiation Formula.
numerical-methods-mechanismNotes
Backward Differentiation Formula 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 II
BookReference text covering stiff ODE solvers and backward differentiation formulas.
Explore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageAlternative toAdams-Bashforth Method
BDF methods are implicit multistep methods for stiff problems.
Depends onImplicit Integration
BDF methods require solving implicit equations at each step.
Related Algorithms
Tags
Numerical MethodsOde SolverImplicit MethodStiff System