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Scientific Computing. Numerical Solution Of Ordinary Differential Equations - Euler’s Method. Bungee Jumper – Coupled ODEs. Bungee Jumper – Coupled ODEs. We want to model the vertical dynamics of a jumper connected to a stationary platform with a bungee cord. F=ma Forces :

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scientific computing
Scientific Computing

Numerical Solution Of

Ordinary Differential Equations

- Euler’s Method

bungee jumper coupled odes1
Bungee Jumper – Coupled ODEs
  • We want to model the vertical dynamics of a jumper connected to a stationary platform with a bungee cord. F=ma
  • Forces:
    • mg (gravity, g = acceleration due to gravity)
    • cd v2(drag force, cd = drag coefficient, v = velocity)

(need to always retard v, so if falling (v>0) need force neg,

if rising (v<0) need force pos to reduce dv/dt)

(use sign(v) for drag force)

    • k (x-L) (spring force, x = distance measured down from

platform, L = rest length of cord)

    • γ v (damping force, γ is damping coefficient of cord)
bungee jumper coupled odes2
Bungee Jumper – Coupled ODEs
  • We get
  • Dividing by m we get:

k =  = 0 if x  L

k =  = 0 if x  L

bungee jumper coupled odes3
Bungee Jumper – Coupled ODEs
  • Need to solve two simultaneous ODEs for x and v

k =  = 0 if x  L

solving first order ode s
Solving First Order ODE’s
  • A general form for a first order ODE is
  • Or alternatively
  • We desire a solution x(t) which satisfies this ODE and one specified boundary condition.

x(a) = c

initial value problem
Initial Value Problem
  • An Initial Value Problem (IVP) is to solve a differential equation (of some order) subject to one or more initial conditions.
analytic solutions
Analytic Solutions
  • Suppose
  • This can be clearly solved by integration:
numerical solution
Numerical Solution
  • For many ODE’s integration is not possible, need to do a numerical approximation to the solution.
  • To find a numerical solution we divide the interval [a,b] for the independent variable t into n subintervals or steps.
numerical solution1
Numerical Solution
  • Then the value of the true solution is approximated at these n values of t.
  • We denote the approximation at these pts by so that
numerical solution2
Numerical Solution
  • Let x(t) be the actual solution for dx/dt at the step values.
  • Then, assuming no roundoff error, the difference in the calculated and true value is the truncation error,
numerical solution3
Numerical Solution

Numerical algorithms for solving 1st order odes for an initial condition are based on one of two approaches:

  • Direct or indirect use of the Taylor series expansion for the solution function
  • Use of open or closed integration formulas.
numerical solution4
Numerical Solution

The various solution algorithms are classified into:

  • One-step: calculation of xi+1 given the differential equation and previous value: xi
  • Multi-step: in addition to the previous information the algorithm requires values of t and x outside of the interval under consideration.
numerical solution5
Numerical Solution
  • The first method considered is based on the Taylor series expansion.
  • It forms the basis for some of the other methods.
taylor series
Taylor Series
  • For this method we express the solution x(t) about some starting point t0 using a Taylor expansion.
  • The ODE is given by
taylor series1
Taylor Series
  • Thus, by substitution from
  • Where


… etc

error analysis for taylor approximation
Error Analysis for Taylor Approximation
  • Suppose we use the nth order Taylor series approximation

to generate the value of x1=x(t1). How close are we to the actual value?

error analysis for taylor approximation1
Error Analysis for Taylor Approximation
  • In general, suppose we generate points for x by:

to generate the value of xi+1=x(ti+1). How close are we to the actual value?

error analysis for taylor approximation2
Error Analysis for Taylor Approximation
  • The exact value for x(ti+1) is given by
error analysis for taylor approximation3
Error Analysis for Taylor Approximation
  • Algorithms for which the last term in expansion is dropped are of order hn.
  • The error is of order hn+1.
  • The local truncation error is bounded as follows
  • where
taylor series solution
Taylor Series Solution
  • Note: differentiation of f(t,x) can be complicated.
  • Direct Taylor expansion is not used other than the simplest case,
euler s method1
Euler’s Method
  • The problem with the simple Euler method is the inherent inaccuracy in the formula.
euler s method2










Euler’s Method
  • The geometric interpretation is shown the diagram.
euler s method3










Euler’s Method
  • The solution across the interval [t0,t1] is assumed to follow the line tangent to x(t) at t0.
euler s method4
Euler’s Method
  • When the method is applied repeatedly across several intervals in sequence, the numerical solution traces out a polygon segment with sides of slope fi, i=0,1,2,…,(n-1).
euler s method5
Euler’s Method
  • The simple Euler method is a linear approximation, and is a perfect solution only if the function is linear (or at least linear in the interval).
  • This is inherently inaccurate.
  • Because of this inaccuracy small step sizes are required when using the algorithm.

A ball at 1200K is allowed to cool down in air at an ambient temperature of 300K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by

Find the temperature at

seconds using Euler’s method. Assume a step size of



Step 1:

is the approximate temperature at

solution cont
Solution Cont

Step 2:


is the approximate temperature at

solution cont1
Solution Cont

The exact solution of the ordinary differential equation is given by the solution of a non-linear equation as

The solution to this nonlinear equation at t=480 seconds is

comparison of exact and numerical solutions
Comparison of Exact and Numerical Solutions

Figure 3. Comparing exact and Euler’s method

effect of step size
Effect of step size

Table 1. Temperature at 480 seconds as a function of step size, h


comparison with exact results
Comparison with exact results

Figure 4. Comparison of Euler’s method with exact solution for different step sizes