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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
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.


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


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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