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Scientific Computing. Numerical Solution Of Ordinary Differential Equations - Runge-Kutta Methods. Runge-Kutta Methods. If we take more terms in the Taylor’s Series expansion of f(t,x), we would get a more accurate solution algorithm.

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Scientific Computing

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Scientific computing

Scientific Computing

Numerical Solution Of

Ordinary Differential Equations

- Runge-Kutta Methods


Runge kutta methods

Runge-Kutta Methods

  • If we take more terms in the Taylor’s Series expansion of f(t,x), we would get a more accurate solution algorithm.

  • Problem: It is difficult to compute higher-order derivatives numerically.

  • Idea: Approximate the value of higher derivatives of f(t,x) by evaluating f several times between iterates: ti and ti+1 .


Better methods

Better Methods

  • The ODE +IV x(a)= x0 =s

    can be alternatively solved as an integral equation. For example, on the first interval from x0 to x1 :


Better methods1

Better Methods

In general:

  • The problem is that we don’t know x(t), so evaluating the integral is not trivial.

  • We assume that if h is small enough, f(t,x) will not change that much over the interval [ti, ti+1]. Using “left-rectangular integration,”

  • This is the Euler Method. A family of improved versions are called Runge-Kutta methods


Euler method

Euler Method

Euler Solution

x

xi

true solution

h

ti

ti+1 = ti + h

Tuncation error:

true solution


Euler vs actual solution

Euler vs Actual Solution

Euler Solution

RK methods differ in how they estimate fi

For Euler Method (a type of RK method), fi ≈ f(ti, xi)

x

xi

true solution

fi

h

ti

ti+1 = ti + h


Runge kutta order 2 method

Runge-Kutta Order 2 Method

x

xi

fi

h

Weighted average of two slopes

x

ti

ti+αh

ti+1


Runge kutta order 2 method1

Runge-Kutta Order 2 Method

  • Let

  • Then,

  • Expand k2 using Taylor’s Series in (t,x):

    where is a point between


Runge kutta order 2 method2

Runge-Kutta Order 2 Method

Then,

The actual Taylor Series expansion for x(ti+h) is:

Thus, our weighted average approximation will be good to

O(h3) if


Runge kutta order 2 method3

Runge-Kutta Order 2 Method

So, general RK order 2 Method is:

Examples:

1) ω1=1, ω2=0 -> Euler’s Method (Still O(h2) as no averaging)

Error is lower than Euler, but

now there are 2 calls to f(x,y)

required per step

Truncation error:

true solution


Heun s method pav s rk order 2

x

xi+1

xi

x

ti+1

ti

Heun’s Method (Pav’s RK-Order 2)

Heun’s method

resulting in

where

Heun’s method is ~ Trapezoidal Rule in Integration


Midpoint method

Midpoint Method

resulting in

Midpoint method is ~ Midpoint Rule in Integration

where


Ralston s method

Ralston’s Method

Ralston (1962) and Ralston and Rabinowitiz (1978)

determined that choosing ω2 = 2/3 provides a minimum

bound on the truncation error for the second order RK

algorithms.

resulting in

where


Example

Example

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 Heun’s method. Assume a step size of

seconds.

http://numericalmethods.eng.usf.edu


Solution

Solution

Step 1:

http://numericalmethods.eng.usf.edu


Solution cont

Solution Cont

Step 2:

http://numericalmethods.eng.usf.edu


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

http://numericalmethods.eng.usf.edu


Comparison with exact results

Comparison with exact results

Figure 2. Heun’s method results for different step sizes

http://numericalmethods.eng.usf.edu


Effect of step size

Effect of step size

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

(exact)

http://numericalmethods.eng.usf.edu


Comparison of euler and runge kutta 2 nd order methods

Comparison of Euler and Runge-Kutta 2nd Order Methods

Table 2. Comparison of Euler and the Runge-Kutta methods

(exact)

http://numericalmethods.eng.usf.edu


Comparison of euler and runge kutta 2 nd order methods1

Comparison of Euler and Runge-Kutta 2nd Order Methods

Table 2. Comparison of Euler and the Runge-Kutta methods

(exact)

http://numericalmethods.eng.usf.edu


Comparison of euler and runge kutta 2 nd order methods2

Comparison of Euler and Runge-Kutta 2nd Order Methods

Figure 4. Comparison of Euler and Runge Kutta 2nd order methods with exact results.

http://numericalmethods.eng.usf.edu


Runge kutta order 4 method

Runge-Kutta Order 4 Method

weighted average

where

different

estimates

of slope


Runge kutta order 4 method1

Runge-Kutta Order 4 Method

k2

k4

x

k3

k1

ti

ti + h/2

ti + h


Runge kutta order 4 example

Runge-Kutta Order 4 Example

  • Consider Exact Solution

  • The initial condition is:

  • The step size is:


Runge kutta order 4 example1

Runge-Kutta Order 4 Example

The example of a single step:


Runge kutta order 4 example2

Runge-Kutta Order 4 Example

The RK values are nearly equivalent to those of the exact solution out to x=5.

y(5) = -111.4129 (-111.4132)

The error is small relative to the exact solution.


Runge kutta order 4 example3

Runge-Kutta Order 4 Example

A comparison between the 2nd order and the 4th order Runge-Kutta methods show a slight difference.


Scientific computing

ERRORS

Local

(single-step)

Global

(multi-step)

function

evals per step

RK

1st-order

1

O(h2)

O(h) = O(1/n)

2nd-order

O(h3)

O(h2) = O(1/n2)

2

4th-order

O(h5)

O(h4) = O(1/n4)

4

“local error”

For n steps,

Therefore, “global error”

Order of global error is 1 less than order of local error


Scientific computing

A fair comparison between two RK methods must account

the differing number of function calls per step

RK1 (Euler Method)- - 1 function call per step

t0

t1

t2

t3

t4

t5

t8

t7

t6

RK4 - - 4 function calls per step

t0

t1

t2

Example: integrate from 0 to 4 using n=400 function evaluations

1st-order Euler with h = 0.01… global error = O(0.01)

4th-order RK with h = 0.04… global error = O(0.00000256)

104 times less error for same amount of work  FREE LUNCH !!!


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