Knowledge card
Shooting Method (BVP → IVP)
Mathematical Modelling — Fluid Dynamics & Heat TransferNumerical analysis; boundary-layer similarity theory
Symbol
The conversion of the boundary-value problem into a root-finding problem over initial-value problems: guess the unknown wall values, integrate as an IVP, measure how badly the far-field conditions are missed, and iterate the guesses until the miss vanishes.
Open full derivation chain: 5 stepsDerivation
- 1
Open Step 1: Given / definition · The BVP supplies f(0) = S, f′(0) = 1 + Lf″(0), θ′(0) = −Bi(1 − θ(0)) at the wall and f′ → 0, θ …
The BVP supplies f(0) = S, f′(0) = 1 + Lf″(0), θ′(0) = −Bi(1 − θ(0)) at the wall and f′ → 0, θ → 0 in the far field — five conditions split across two ends.Given / definition Defining ruleThe starting relation of a derivation — the definition or established equation every later step transforms; it justifies no change itself. - 2
Open Step 2: Shooting: guess unknown initial conditions · Guess the missing wall values s₁ = f″(0) and s₂ = θ(0). With them the wall state (f, f′, f″, θ,…
Guess the missing wall values s₁ = f″(0) and s₂ = θ(0). With them the wall state (f, f′, f″, θ, θ′)(0) = (S, 1 + Ls₁, s₁, s₂, −Bi(1 − s₂)) is COMPLETE — the BVP has become an IVP parameterised by s = (s₁, s₂).Shooting: guess unknown initial conditions Defining ruleguess s = (f″(0), θ(0)); solve the IVP; find s with R(s) = 0Converting a boundary-value problem into root-finding over initial-value problems: unknown wall values are guessed, integrated, and corrected until the far-field residual vanishes. - 3
Open Step 3: Advance one integration step · March the first-order system y′ = F(η, y) from η = 0 to η_max with classical RK4 (or any high-o…
March the first-order system y′ = F(η, y) from η = 0 to η_max with classical RK4 (or any high-order one-step method) — each step advances the full five-component state.Advance one integration step Defining ruleyₙ₊₁ = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄)Advancing the state one mesh step with a one-step integrator (classical RK4 shown). - 4
Open Step 4: Iterative correction (root update) · Evaluate the far-field residual R(s) = (f′(η_max; s), θ(η_max; s)) and update the guesses by a …
Evaluate the far-field residual R(s) = (f′(η_max; s), θ(η_max; s)) and update the guesses by a Newton or secant step on R(s) = 0 — each iteration is one full IVP integration per residual component.Iterative correction (root update) Defining rulesₖ₊₁ = sₖ − J⁻¹R(sₖ)Updating the current guess from the residual (Newton / secant correction) until convergence. - ★
Open Step 5: Convergence / tolerance acceptance · Accept when ‖R(s)‖ falls below tolerance (e.g. 10⁻⁶): the converged s₁ IS f″(0) (the skin-frict…
Accept when ‖R(s)‖ falls below tolerance (e.g. 10⁻⁶): the converged s₁ IS f″(0) (the skin-friction input), and θ′(0) follows from the convective wall condition — the BVP solution has been recovered from IVPs.Convergence / tolerance acceptance Defining rule‖R‖ < tol accepted; halve h (or double η_max) and compareAccepting a numerical solution only when the residual is below tolerance and refinement no longer changes the answer.
Equation Workspace
Set known values, calculate unknowns, inspect dependencies, and run numerical solves.
Calculator
Equation Workspace
Set known values, calculate unknowns, inspect dependencies, and run numerical solves.
Equation workspace
Set the values you know, then calculate the remaining quantities in Shooting Method (BVP → IVP).
Domain Analysis
Unit status, dimensional trails, per-step checks, and custom unit expressions.
Domain Analysis
Unit status, dimensional trails, per-step checks, and custom unit expressions.
Step-by-step unit check (5 steps)
Step 1 given
Expand checked this step.Step 2 shooting_guess
Expand checked this step.Step 3 integrate_step
Expand checked this step.Step 4 iterate_correction
Expand checked this step.Step 5 convergence_check
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: s→R(s)
“s→R(s)” 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.
Mathematical Analysis
Object type, lawful operations, conditions, comparisons, and derivation-step checks.
Check one step at a time (5 steps)
Step 1 — given
Step 2 — shooting guess
Step 3 — integrate step
Step 4 — iterate correction
Step 5 — convergence check
Essence
S1Guessing s = (f″(0), θ(0)) completes the wall state: the boundary-value problem becomes an initial-value problem in s.
S2The IVP is integrated across the truncated domain by fourth-order Runge–Kutta marching.
S3The far-field miss R(s) drives a Newton/secant update of the guessed slopes — shooting is root-finding over integrations.
Dimension I
Definition
What it is — and what it is not
- What it is
- S4A BVP-to-IVP conversion: complete the wall state with guessed parameters s = (f″(0), θ(0)), integrate, and drive the far-field residual R(s) to zero by Newton/secant iteration. [1.1]
- S5The two-point problem: three wall conditions plus two far-field decays — no end carries a complete initial state.
- Wisdoms
- S6At convergence the shooting solution satisfies every boundary condition: f″(0) and θ′(0) — the wall quantities the study reports — are read directly from the converged state.
Dimension II
In practice
How to deal with it
- How to deal with it
- S7Bracket s from physics before iterating: f″(0) < 0 for a stretching wall (the wall drags the fluid) and 0 < θ(0) ≤ 1 under convective heating. A bvp4c solution supplies an excellent starting guess.
- Warnings
- S8For growing modes the IVP can be exponentially sensitive to s (the classical shooting instability); if iterates diverge, shorten η_max, use multiple shooting, or fall back to collocation. [1.1]