Mathematical Modelling Lecture 11 Differential …pjh503/mathematical_model/math...Title Mathematical Modelling Lecture 11 Differential Equations Author Phil [email protected]

IntroductionDifferential equations

Euler methodError analysis

Mathematical ModellingLecture 11 – Differential Equations

Phil [email protected]

Phil Hasnip Mathematical Modelling



Overview of Course

Model construction −→ dimensional analysisExperimental input −→ fittingFinding a ‘best’ answer −→ optimisationTools for constructing and manipulating models −→networks, differential equations, integrationTools for constructing and simulating models −→randomnessReal world difficulties −→ chaos and fractals

A First Course in Mathematical Modeling by Giordano, Weir &Fox, pub. Brooks/Cole. Today we’re in chapter 10.




Change?!

We often have to model dynamic systems.

Discrete −→ difference equationsContinuous −→ differential equations

Today we’re looking at differential equations.




Change?!

We often have to model dynamic systems.

Discrete −→ difference equationsContinuous −→ differential equations

Today we’re looking at differential equations.




Differential equations

The basic problem:

dxdt

= x = f (x , t)

subject to given boundary conditions.

Of course we may be interested in higher-order differentialequations!




Initial value problems

If the boundary conditions are all expressed at the same point,then we have an initial value problem.

x = f (x , t)

⇒ x (t) = x (0) +

∫ t

0f(x , t ′

)dt ′

NB extrapolation −→ can lead to large errors.

How do we solve this?




General method

First we split integral into discrete blocks:

x (t) = x (0) +

∫ ∆t

0f(x , t ′

)dt ′ +

∫ 2∆t

∆tf(x , t ′

)dt ′ +

. . .+

∫ (n+1)∆t

n∆tf(x , t ′

)dt ′ + . . .+

∫ t

t−∆tf(x , t ′

)dt ′

Write each term f (x , t) as Taylor series about lower limit

f (x , t) = f (x ,n∆t) + (t − n∆t)∂f∂t

∣∣∣∣t=n∆t

+(t − n∆t)2

2∂2f∂t2

∣∣∣∣t=n∆t

+ . . .




General method

First we split integral into discrete blocks:

x (t) = x (0) +

∫ ∆t

0f(x , t ′

)dt ′ +

∫ 2∆t

∆tf(x , t ′

)dt ′ +

. . .+

∫ (n+1)∆t

n∆tf(x , t ′

)dt ′ + . . .+

∫ t

t−∆tf(x , t ′

)dt ′

Write each term f (x , t) as Taylor series about lower limit

f (x , t) = f (x ,n∆t) + (t − n∆t)∂f∂t

∣∣∣∣t=n∆t

+(t − n∆t)2

2∂2f∂t2

∣∣∣∣t=n∆t

+ . . .




Euler method

The Euler method assumes f varies slowly over the range

n∆t ≤ t < (n + 1) ∆t

Thus we can ignore most of these terms:

f (x , t) = f (x ,n∆t) + (t − n∆t)∂f∂t

∣∣∣∣t=n∆t

+ . . .

≈ f (x ,n∆t)

i.e. replace the varying function in each integral by the functionvalue at the start.




Euler method

We can now write:∫ (n+1)∆t

n∆tf(x , t ′

)dt ′ ≈

∫ (n+1)∆t

n∆tf (x (n∆t) ,n∆t) dt ′

= f (x (n∆t) ,n∆t) ∆t




Example 1: radioactive decay

dxdt

= −λx

which we can solve analytically. However, let’s use Euler –f (x , t) = −λx so we have:

xn+1 = xn + (−λxn)∆t

Let’s see what happens for λ = 1 and ∆t = 0.1.





λ = 1, ∆t = 0.1:

n tn xn analytic0 0.0 1.0 1.01 0.1 0.90 0.9048372 0.2 0.81 0.818731...

......

...10 1.0 0.348678 0.367879...

......

...50 5.0 0.005154 0.006738





The error doesn’t seem too bad.

Can reduce error by reducing ∆t −→ more stepsThis error is called truncation errorCan we keep reducing error this way? No!Also have finite precision, or roundoff errorthis limits the final accuracy of any numerical scheme




Example 2: simple harmonic oscillator

Now we’ll look at a second-order differential equation:

d2xdt2 = −ω2x

We’ll split this into two coupled first order equations:

dxdt

= v

dvdt

= −ω2x

so using the Euler method we have:

xn+1 = xn + vn∆tvn+1 = vn − ω2xn∆t





Typical boundary conditions: x(0) = 1; v(0) = 0. Let’s useω = 1 and ∆t = 0.1 as before:

n tn xn analytic x vn analytic v0 0.0 1.0 1.0 0.0 0.01 0.1 1.0 0.995004 -0.1 -0.0998332 0.2 0.99 0.980067 -0.2 -0.198669...

......

......

...10 1.0 0.570790 0.540302 -0.882508 -0.841471...

......

......

...50 5.0 0.343355 0.283662 1.235613 0.958924





Typical boundary conditions: x(0) = 1; v(0) = 0. Let’s useω = 1 and ∆t = 0.1 as before:

n tn xn analytic x vn analytic v0 0.0 1.0 1.0 0.0 0.01 0.1 1.0 0.995004 -0.1 -0.0998332 0.2 0.99 0.980067 -0.2 -0.198669...

......

......

...10 1.0 0.570790 0.540302 -0.882508 -0.841471...

......

......

...50 5.0 0.343355 0.283662 1.235613 0.958924





Our Euler method leads to a nonphysical velocity v !Clearly something has gone wrong.Why?




Error analysis for Euler method

Let’s measure the error at step n, εn, by the residual:

εn = x truen − xn

⇒ x truen = xn + εn

Now according to Euler,

x truen+1 = x true

n + f(x true

n , t)

∆t⇒ (xn+1 + εn+1) = (xn + εn) + f (xn + εn, t) ∆t





Assuming εn is small, we can use a Taylor expansion in x toshow that:

εn+1 = εn

(1 +

∂f∂x

∣∣∣∣x=xn

∆t

)= εnG

where G is the error gain term.





Provided |G| < 1 the error term does not grow, so we have astable scheme.

E.g. for radioactive decay we have f (x , t) = dxdt = −λx , so

G = 1 +∂f∂x

∣∣∣∣x=xn

∆t

= 1− λ∆t

Thus the Euler method is stable if |1− λ∆t | < 1, i.e.

0 < ∆t <2λ




Beyond Euler

Of course we can do better than Euler:

Predictor-correctorHigher-order methods (extending truncation scheme)




Beyond Euler

These schemes generally:

Are more accurate and stableAllow larger ∆tRequire more function evaluations per step and morecomputer storage

One common scheme is 4th-order Runge-Kutta – see any texton numerical methods for lots of details!




Summary

The Euler method can solve differential equationsnumericallyCare must be taken when choosing the time-stepHigher-order methods may be required


Mathematical Modelling Lecture 11 Differential …pjh503/mathematical_model/math...Title Mathematical Modelling Lecture 11 Differential Equations Author Phil [email protected]

Documents