Knowledge card

BDF — Backward Differentiation Formula (shooting)

Symbol

BDF2

A single-shooting boundary-value method: it marches an initial-value guess across the interval with an implicit second-order backward-differentiation (BDF2) integrator and corrects the unknown initial slopes by damped Newton iteration until the terminal boundary residuals vanish.

Main equation

Level 0 of 1

Brief — named quantities

About levels

Why the levels load one at a time

Only level 0 travels with this article; the expansion runs to level 1. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them.

To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.

(((\frac{3}{2} y_{n + 1} S2.1 - 2 y_{n}) S2.2 + \frac{1}{2} y_{n - 1}) S2.3 = h \cdot f (η_{n + 1}, y_{n + 1}))

Dimension:1Maths:𝔹 equation — A statement that two sides are equalDimensionally verifiedMathematically verifiedFully reduced to base quantities

In words:3/2 times minus2 times plus1/2 times equals.

Open full derivation chain: 2 stepsDerivation

1
Open Step 1: Given / definition · The startup row, and the only formula available at the very first step: with no earlier state t…
The startup row, and the only formula available at the very first step: with no earlier state to differentiate through, the backward difference is the plain one-step difference. This is a first-order implicit rule — the right-hand side is still evaluated at the NEW point, which is where the stability comes from.Given / definition
Level 0 · more available
Brief — named quantities
About levels
Why the levels load one at a time
Only level 0 travels with this article. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them — and how deep it goes is not known until you get there.
To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.
How many levels remain is not yet known — the depth is discovered one level at a time.
$((y_{n + 1} S1.1 - y_{n}) S1.2 = h \cdot f (η_{n + 1}, y_{n + 1}))$
★
Open Step 2: Advance one integration step · Once one step has been taken a second earlier state exists, so the backward difference can be f…
Once one step has been taken a second earlier state exists, so the backward difference can be fitted through BOTH previous points instead of one — raising the formula to order 2, with the authored coefficients ((3/2), 2, (1/2), signed) as its weights. This is the formula the integrator marches with for every step after the first. Because the right-hand side is evaluated at the NEW point it depends on the unknown y_{n+1}, so each step solves an inner nonlinear system — the price of the stability an implicit backward formula buys.Advance one integration step
Open term-change ledger: 1 records
Open record 1: ∅ → y_{n-1}
Consumes:∅Produces:L0 $y_{n - 1}$ Admitting the second backward point y_{n-1} into the stencil is what raises the order from 1 to 2; the authored coefficients are the unique weights that make the backward difference match the derivative at the new point to that order.y_{n-1}
Level 0 · more available
Brief — named quantities
About levels
Why the levels load one at a time
Only level 0 travels with this article. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them — and how deep it goes is not known until you get there.
To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.
How many levels remain is not yet known — the depth is discovered one level at a time.
$(((\frac{3}{2} y_{n + 1} S2.1 - 2 y_{n}) S2.2 + \frac{1}{2} y_{n - 1}) S2.3 = h \cdot f (η_{n + 1}, y_{n + 1}))$

Equation Workspace

Set known values, calculate unknowns, inspect dependencies, and run numerical solves.

Calculator

Equation workspace

Set the values you know, then calculate the remaining quantities in BDF — Backward Differentiation Formula (shooting).

Domain Analysis

Unit status, dimensional trails, per-step checks, and custom unit expressions.

Step-by-step unit check (2 steps)

Step 1 given
Expand checked this step.
Step 2 integrate_step
Expand checked this step.

Try your own — the unit calculator

Ever wondered whether a formula “adds up”? This tool answers one plain question: do the units match? Pick a couple of quantities (distance, time, a speed…), join them with ×, ÷, +, or =, and press Run. It works out the resulting unit — for example distance ÷ time gives a speed, LT⁻¹ (metres per second) — and flags anything that can’t be right, like adding a length to a time.

Reads as: BDF2

“BDF2” is analysed by expanding its full defining equation (its children and their relations), not as a bare symbol. Use Check this concept’s units above for the same result.

Mathematical Analysis

Object type, lawful operations, conditions, comparisons, and derivation-step checks.

Check one step at a time (2 steps)

Step 1 — given

Step 2 — integrate step

Essence
S1The startup row, and the only formula available at the very first step: with no earlier state to differentiate through, the backward difference is the plain one-step difference. This is a first-order implicit rule — the right-hand side is still evaluated at the NEW point, which is where the stability comes from.
S1.1essencey_{n+1} reduced by y_n.
S1.2essence(y_{n+1} − y_n) set equal to h·f(η_{n+1}, y_{n+1}).
S2Once one step has been taken a second earlier state exists, so the backward difference can be fitted through BOTH previous points instead of one — raising the formula to order 2, with the authored coefficients ((3/2), 2, (1/2), signed) as its weights. This is the formula the integrator marches with for every step after the first. Because the right-hand side is evaluated at the NEW point it depends on the unknown y_{n+1}, so each step solves an inner nonlinear system — the price of the stability an implicit backward formula buys.
S2.1essencey_{n+1} reduced by y_n.
S2.2essenceThe sum of (y_{n+1} − y_n) and y_{n-1}.
S2.3essence((y_{n+1} − y_n) + y_{n-1}) set equal to h·f(η_{n+1}, y_{n+1}).

Dimension I

Definition

What it is — and what it is not

Dimension II

In practice

How to deal with it

No practical guidance recorded yet.

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Encyclopedia

BDF — Backward Differentiation Formula (shooting)

Equation Workspace

Domain Analysis

Mathematical Analysis