Numerical Methods - Prof. Vergara

Name: Numerical Methods - Prof. Vergara
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Revisionato il 17/05/2026

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Appunti chiari e completi dell'intero corso di "Numerical methods" (esercitazioni incluse) tenuto dal Prof. Vergara basati su appunti personali del publisher presi alle lezioni del professore …

Esame Mathematical and numerical methods in engineering

Facoltà Ingegneria

Dal corso del Prof. Vergara Christian

Università Università degli Studi di Pisa

A.A. 2016-2017

134 pagine

Appunto

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Estratto del documento

NUMERICAL METHODS

DISCRETIZZAZIONE DELLE EQUAZIONI DIFFERENZIALI ORDINARIE

PARTIAL DIFFERENTIAL EQUATION

REAL PHENOMENON

We choose a model (this is a choice driven by what we want to see)

BLOOD IN ARTERIES

There are lots of continuous models that can be chosen.

PDE

We choose a discretization model like NAVIER-STOKES

DISCRETIZED PROBLEM

X ← MODELING ERROR → X_M NUMERICAL ERROR → X_h

IMPLEMENT AN ALGORITHM

CODE

We obtain FINITE ELEMENTS

FEM code

Also in this case we can choose several implementations

ROUND-OFF ERROR (errore di arrotondamento)

It's important that the discretization doesn't change our model.We have to quantify the error made when we choose a PDE to model a real phenomenon.We don't know X_h and so this generates the need to estimate the error.The numerical analysis transforms the PDE into a discretized problem.Using the computer we are introducing other errors. The discretized problem is correctly solved only by hand, while using the PC we made a ROUND-OFF ERROR.

NUMERICAL METHODS

DISCRETIZZAZIONE DELLE EQUAZIONI DIFFERENZIALI ORDINARIE

PARTIAL DIFFERENTIAL EQUATION

REAL PHENOMENON → we choose a model (this is a choice driven by what we want to see) → PDE → we choose a discretization model like NAVIER-STOKES → DISCRETIZED PROBLEM

There are lots of continuous models that can be chosen.

X ← MODELING ERROR → X_M ← NUMERICAL ERROR → X_h

IMPLEMENT OUR ALGORITHM → CODE

We obtain FINITE ELEMENTS → FEM code

also in this case we can choose Several implementations

ROUND-OFF ERROR (errore di arrotondamento)

It's important that the discretization doesn't change our model.

We have to quantify the error made when we choose a PDE to model a real phenomenon.

We don't know X_h and so this generates the need to estimate the error.

The numerical analysis transforms the PDE into a discretized problem.

Using the computer we are introducing other errors. The discretized problem is correctly solved only by hand, while using the PC we made a ROUND-OFF ERROR.

Approximation of Derivatives

Let x_j, for some j's, be points in ℝ

and suppose h = x_j - x_j-1

where the distance is independent by j

I know the values of a function ν(x) in these points, so we know

ν(x_j-1), ν(x_j),...

How could I approximate νʹ(x_j) = ?

Example:

Forward Euler Formula: νʹ(x_j) ≃ D₊ν(x_j)
νʹ(x_j) ≃ (ν(x_j+1) - ν(x_j)) / h

This is an easy way to approximate derivatives

The formula is convergent and is of the first order.

We introduce the error:

Ε_j = |νʹ(x_j) - D₊ν(x_j)| = |νʹ(x_j) - (ν(x_j+1) - ν(x_j)) / h| = O(h)

Taylor Expansion

ν(x) = ν(y) + νʹ(y)(x-y) + νʺ(y)/2 (x-y)² + νʺʹ(y)/6(x-y)³ + Θ(|x-y|⁴)

A quantity q is said to be O(ε) if lim ε/ε ⁶ < ∞ ⋆

Take x = x_j+1, y = x_j; ν(x_j+1) = ν(x_j) + νʹ(x_j)h + Θ(h²)

Q. ∋: h > 0 hence h = x-y where x = x_j+1 & y = x_j

⇒ Θ(h²) = -ν(x_j+1) - ν(x_j) + νʹ(x_j)h ≫ Θ(h) = νʹ(x_j)

RICORDIAMO che IL SEGNO DI Θ(h) è INDIFFERENTE

If I have a ratio between 2 numbers and the denominator goes to 0, I have 3 possibilities:

the _N/_D → ∞ when l goes to 0 faster than q
the _N/_D number ≠ 0 l ≃ 0 with the same velocity
the _N/_D → 0 q goes to 0 faster than l

h is a parameter which characterizes my discretization. The more the h is smaller and the more my discretization is precise.

In general, given an approximation D_j, the formula to approximate the derivative is said to be correct if:

|E_j = |D_j-v'(x_j)| = θ(h^p)

The errors become 0 when the distance between 2 consequent points become 0.

In particular, the formula is of order p. In this way, I’m also saying the velocity of convergence, This p allows us to compare several methods,

p can be 1, 2 or other numbers

The fast one is 2 but it has also some negative facts.

• Backward Euler Formula

v'(x_j) ≃ D_-v(x_j) =

(v(x_j) - v(x_j-1)) / h

Accuracy of this formula:

We can write Taylor expansion:

v(x_j-1) = v(x_j) - v'(x_j) h +

Anteprima

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