Numerical Analysis of Differential Equations Initial Value Problems (II)

Fakult¨at f¨ur Mathematik und Informatik 8. Mai 2012 TU Bergakademie Freiberg

W. Queck/M. Helm

Numerical Analysis of Differential Equations Initial Value Problems (II)

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Part A – to be prepared before the date of problem session

Exercise 1

Determine an approximate solution for the initial value problem y⁰=−(y+ 1)(y+ 3), 0≤t≤2 y(0) =−2

applying

a) the explicitEulerscheme, b) the modifiedEulermethod, c) the improvedEuler method

with step sizeh= 0.5 andh= 0.2. Compare your approximations with the corresponding values of the exact solutiony(t) =−3 + 2(1 +e^−2t)⁻¹.

Exercise 2

Rewrite each of the followingButcher/Runge-Kuttatableaus as a set of explicit formulae for the calculation ofyj+1. Do they describe explicit or implicit methods?

a) four-stageEnglandformula 0

1/2 1/2 1/2 1/4 1/4

1 0 -1 2

1/6 0 2/3 1/6 b) Butcher’s formula

0

1/8 1/8

1/4 0 1/4

1/2 1/2 -1 1

3/4 3/16 0 0 9/16

1 -5/7 4/7 12/7 -12/7 8/7

7/90 0 32/90 12/90 32/90 7/90 c) (one)RadauIIA method

1/3 5/12 -1/12

1 3/4 1/4

3/4 1/4

Initial Value Problems 2

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Part B – Main Part

Exercise 3

Find an approximate solution to the initial value problem y⁰(x) =−1000(y−e^−x)−e^−x y(0) = 1

with step sizeh= 0.1 by application of the a) explicitEulermethod,

b) implicitEulermethod.

Compare the approximations forx= 0.1, x= 0.5 andx= 1.0 with the exact solutiony(x) =e^−x. Exercise 4

For the solutiony(x) of the initial value problem

y⁰=−(y+ 1)(y+ 3), 0≤x≤1 y(0) =−2

find an approximation ofy(1) using the classical Runge–Kuttamethod with step size h= 0.5.

Exercise 5

Determine an approximate solution to the following system of ODEs

˙ y(t) =

6 −3

2 1

y(t), y(0) = 5

3

, 0≤t≤1 Therefor apply

a) the explicitEulermethod with step sizeh= 0.25, b) Heun’s method with step size h= 0.5.

Compare the approximations with the exact solution y(t) =

exp(3t) 3 exp(4t) exp(3t) 2 exp(4t)

−1 2

.

Exercise 6

Consider the following initial value problem fory(t).

y⁰⁰⁰ =1

2y², t≥1

y(1) = 1, y⁰(1) =−1, y⁰⁰(1) = 0.5

Determine approximations toy(2),y⁰(2) undy⁰⁰(2) by application of the explicit and the implicit Eulermethod. Take a step size ofh= 1 in both cases.

Initial Value Problems 3 Exercise 7

Check the consistency order of Heun’s method by verification of the order conditions (for explicit Runge-Kuttamethods up to order three) known from the lecture.

0 1/3 1/3

2/3 0 2/3 0

1/4 0 3/4

Exercise 8

The following tableau definesRunge’s method. Show that this method possesses consistency order three by checking the necessary conditions for RKM for l = 1, k = 3 and the expansion of the local truncation error in powers ofh.

0 1/2 1/2

1 0 1

1 0 0 1

1/6 2/3 0 1/6