Knowledge card

Collocation BVP Method — Newton Relaxation on an Adaptive Mesh

Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory

Solving the coupled nonlinear boundary-value problem by driving a collocation residual to zero on every mesh interval at once (Hermite–Simpson / 3-stage Lobatto IIIa, fourth order), refining the mesh where a discretization-error indicator is largest. It is the same METHOD FAMILY as MATLABʼs bvp4c/bvp5c and SciPyʼs solve_bvp, which are the references it is compared against — it is not those solvers and does not inherit their error control or published accuracy.

Main equation

(((y_{i + 1} S3.1 - y_{i}) S3.5 - (\frac{h}{6} S3.4 (((F (η_{i}, y_{i}) S3.2 + 4 F (\frac{η _{i} + h}{2 , y _{m}})) S3.3 + F (η_{i + 1}, y_{i + 1}))))) S3.6 = 0)

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

In words:minusminustimes(plus4 times plus)equalszero.

Open full derivation chain: 3 stepsDerivation

1
Open Step 1: Given / definition · Across one interval of the mesh the state advances from y_i to y_{i+1}. That change is exactly …
Across one interval of the mesh the state advances from y_i to y_{i+1}. That change is exactly the integral of the generic right-hand side over the interval — the quantity any discretisation of this family has to approximate, and the only thing the method assumes about the problem.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.
$((y_{i + 1} S1.1 - y_{i}) S1.2 = \int F (η, y) d η)$
2
Open Step 2: Numerical discretisation preparation · Replace the exact integral by sampling the right-hand side at THREE stages — both ends of the i…
Replace the exact integral by sampling the right-hand side at THREE stages — both ends of the interval and its midpoint — weighted (1, 4, 1)/6. The midpoint state is not a new unknown: it is read off the cubic Hermite interpolant through the two endpoints and their slopes, which is exactly what lifts this rule to order 4 where sampling the ends alone reaches only order 2. Both the right-hand and the midpoint samples depend on the unknown y_{i+1}, so the relation is implicit.Numerical discretisation preparation
Open term-change ledger: 1 records
Open record 1: ∫ F(η, y) dη → h/6 + F(η_i, y_i) + F(η_i + h/2, y_m) + F(η_{i+…
Consumes:L0 $\int F (η, y) d η$ Produces:L0 $h /6$ L0 $F (η_{i}, y_{i})$ L0 $F (η_{i} + h /2, y_{m})$ L0 $F (η_{i + 1}, y_{i + 1})$ The exact integral is unavailable in closed form for a general right-hand side, so it is replaced by a three-stage Lobatto IIIa quadrature: F(η_i, y_i), the interior sample F(η_i + h/2, y_m) taken at the Hermite-interpolated midpoint, and F(η_{i+1}, y_{i+1}), scaled by the weight h/6. This is the one approximation the method makes; everything after it is exact algebra.
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.
$((y_{i + 1} S2.1 - y_{i}) S2.5 = (\frac{h}{6} S2.4 (((F (η_{i}, y_{i}) S2.2 + 4 F (\frac{η _{i} + h}{2 , y _{m}})) S2.3 + F (η_{i + 1}, y_{i + 1})))))$
★
Open Step 3: Collocation (piecewise-polynomial residual) · Collect every term on one side: the interval's COLLOCATION RESIDUAL. A whole-mesh method does n…
Collect every term on one side: the interval's COLLOCATION RESIDUAL. A whole-mesh method does not march this forward — it writes one such residual for every interval, adds the boundary conditions, and drives the entire set to zero simultaneously by Newton relaxation. That is why it converges where a forward march diverges.Collocation (piecewise-polynomial residual)
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.
$(((y_{i + 1} S3.1 - y_{i}) S3.5 - (\frac{h}{6} S3.4 (((F (η_{i}, y_{i}) S3.2 + 4 F (\frac{η _{i} + h}{2 , y _{m}})) S3.3 + F (η_{i + 1}, y_{i + 1}))))) S3.6 = 0)$

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 Collocation BVP Method — Newton Relaxation on an Adaptive Mesh.

Domain Analysis

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

Step-by-step unit check (3 steps)

Step 1 given
Expand checked this step.
Step 2 discretise
Expand checked this step.
Step 3 collocation
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: Collocation BVP Method — Newton Relaxation on an Adaptive Mesh

“Collocation BVP Method — Newton Relaxation on an Adaptive Mesh” 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 (3 steps)

Step 1 — given

Step 2 — discretise

Step 3 — collocation

Essence
S1Across one interval of the mesh the state advances from y_i to y_{i+1}. That change is exactly the integral of the generic right-hand side over the interval — the quantity any discretisation of this family has to approximate, and the only thing the method assumes about the problem.
S1.1essencey_{i+1} reduced by y_i.
S1.2essence(y_{i+1} − y_i) set equal to ∫ F(η, y) dη.
S2Replace the exact integral by sampling the right-hand side at THREE stages — both ends of the interval and its midpoint — weighted (1, 4, 1)/6. The midpoint state is not a new unknown: it is read off the cubic Hermite interpolant through the two endpoints and their slopes, which is exactly what lifts this rule to order 4 where sampling the ends alone reaches only order 2. Both the right-hand and the midpoint samples depend on the unknown y_{i+1}, so the relation is implicit.
S2.1essencey_{i+1} reduced by y_i.
S2.2essenceThe sum of F(η_i, y_i) and F(η_i + h/2, y_m).
S2.3essenceThe sum of (F(η_i, y_i) + F(η_i + h/2, y_m)) and F(η_{i+1}, y_{i+1}).
S2.4essenceThe product of h/6 and ((F(η_i, y_i) + F(η_i + h/2, y_m)) + F(η_{i+1}, y_{i+1})).
S2.5essence(y_{i+1} − y_i) set equal to (h/6 · ((F(η_i, y_i) + F(η_i + h/2, y_m)) + F(η_{i+1}, y_{i+1}))).
S3Collect every term on one side: the interval's COLLOCATION RESIDUAL. A whole-mesh method does not march this forward — it writes one such residual for every interval, adds the boundary conditions, and drives the entire set to zero simultaneously by Newton relaxation. That is why it converges where a forward march diverges.
S3.1essencey_{i+1} reduced by y_i.
S3.2essenceThe sum of F(η_i, y_i) and F(η_i + h/2, y_m).
S3.3essenceThe sum of (F(η_i, y_i) + F(η_i + h/2, y_m)) and F(η_{i+1}, y_{i+1}).
S3.4essenceThe product of h/6 and ((F(η_i, y_i) + F(η_i + h/2, y_m)) + F(η_{i+1}, y_{i+1})).
S3.5essence(y_{i+1} − y_i) reduced by (h/6 · ((F(η_i, y_i) + F(η_i + h/2, y_m)) + F(η_{i+1}, y_{i+1}))).
S3.6essence((y_{i+1} − y_i) − (h/6 · ((F(η_i, y_i) + F(η_i + h/2, y_m)) + F(η_{i+1}, y_{i+1})))) set equal to 0.

Dimension I

Definition

What it is — and what it is not

What it is: S4The numerical strategy for the coupled nonlinear BVP.
Attributes: S5MATLAB bvp4c (collocation) cross-checked with the backward differentiation formula (BDF) for stiff stability. [1.1]

Dimension II

In practice

How to deal with it

What to do: S6Validate the computed skin friction and Nusselt number against previously published results.

vproductionmain @ 5d8445a3

Encyclopedia

Collocation BVP Method — Newton Relaxation on an Adaptive Mesh

Equation Workspace

Domain Analysis

Mathematical Analysis