Runge Kutta Method, Lecture Notes - Mathematics, Study notes for Calculus I. University of California (CA) - UCLA

Calculus I

Description: Ranga kutta method, second order, computation solution, fourth order, formulation, multi steps, taylor expand the local truncation error
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Runge-Kutta Method
Adrian Down
April 25, 2006
1 Second-order Runge-Kutta method
1.1 Review
Last time, we began to develop methods to obtain approximate solutions to
the differential equation ˙x=f(t, x(t)) with error of O(h2), where his the
spacing of the mesh used to calculate the approximation. To construct these
methods, our goal was to minimize the local truncation error. We saw that,
in general, the global truncation error should be one order less in h.
Last time, we introduced an approximation that generalized the Modified
Euler method. The general form of the expression was,
y(t+h)y(t) = ω1hF (t) + ω2hf (t+αh, y(t) + βhF (t))
where F(t)f(t, y(t)). Our proposal was to choose ω1, ω2, α and βsuch
that the local truncation error of the approximation is O(h3), from which we
expect the desired global truncation error to be O(h2).
1.2 Computation
We began to evaluate this condition last time by Taylor expanding the func-
tion fup to second order in h. Since fis a function of two variables, we
used the Taylor formula for multiple dimensions. We obtained,
x(t+h)x(t) = h(ω1+ω2)F(t) + h2ω2(αf1?βF fx) + O(h3)
where subscripts indicate partial differentiation. The partial derivatives are
to be evaluated at (t, x(t)).
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The Taylor series of the left side is easily computed,
x(t+h)x(t) = h˙x(t) + h2
2¨x(t) + O(h3)
Since these two expression are equal, match terms in hand cancel coef-
ficients, such that only terms of O(h3) remain. Matching terms in hyields
two equations,
(ω1+ω2)F(t) = ˙x(t)ω2(αft+βF fx) = 1
2¨x(t)
The first equation can be simplified using the definition of f,
F(t) = f(t, x(t)) = ˙x
(ω1+ω2) = 1
The second equation can be solved by taking the time derivative of the
differential equation. This yields partial derivatives, which can be compared
with those on the left of the equation,
˙x=f(t, x(t)) ¨x=ft+fx˙x
=ft+F fx
ω2αft+ω2βF fx=1
2ft+1
2F fx
This equation must hold for all tand x, so it must be that the coefficients of
the partial derivatives are separately equal,
ω2α=ω2β=1
2
1.3 Solutions
We now have three equations and four unknowns. This system is under-
determined, meaning that we should expect a family of solutions in one
parameter.
One possible choice of parameters is,
ω1=1
2ω2=1
2α= 1 β= 1
2
This choice corresponds to the trapezoid rule approximation scheme.
Another possible choice of parameters is,
ω1= 0 ω2= 1 α=1
2β=1
2
This choice corresponds to the Modified Euler method developed earlier.
2 Fourth-order Runge-Kutta method
2.1 Motivation
The fourth-order Runge-Kutta method is commonly used in science and en-
gineering applications. However, the computations required are less than
optimal, as we will see.
2.2 Formulation
The approximation scheme for the fourth-order Runge-Kutta approximation
is,
y(t+h)y(t) = 1
6{F1+ 2F2+ 2F3+F4}
where the Fi’s are recursively defined below.
Note. The Fi’s are constructed such that FiF(t, y(t)) as h0. The
factor of 1
6is to normalize due to the four factors of F. at t= 0.
The definition of the Fi’s can be understood intuitively as attempts to
evaluate the function fat the midpoint and endpoints of the mesh interval
over which the function is being approximated. F1is the value of the function
fat the left endpoint of the interval,
F1=f(t, y(t))
In the spirit of the Modified Euler method, F2is analogous to an attempt at
evaluating the function fat the midpoint of the approximation interval,
F2=ft+h
2, y(t) + h
2F1
3
F3is analogous to a second more accurate attempt to evaluate fat the
midpoint of the interval,
F3=ft+h
2, y(t) + h
2F2
F4is an attempt to evaluate fat the right endpoint of the interval,
F4=f(t, f(t) + hF3)
Note. The Fourth-Order Runge-Kutta method can be used in the case that
fis a vector function provided that fis not a function of time. In this case
fis a function only of position. This is called the autonomous case.
3 Multi-step methods
3.1 Motivation
The fourth-order Runge-Kutta method is not computationally optimal be-
cause the function fmust be evaluated four times to calculate the approxima-
tion at each mesh point. Computations could be prohibitive if the function
fis complicated.
Multistep methods attempt to avoid this problem by creating an approx-
imation based on the values of the approximation at previous points. Al-
though such methods reduce computations, some such methods may not be
stable in all situations. When using multi-step methods, it is necessary to
verify the convergence of the method.
3.2 Two-step method
3.2.1 Setup
We find an explicit two-step method for obtaining an approximation at the
mesh point t+hbased only on the values of the approximation and the
function Fat the previous two mesh points tand th. Forming a general
linear combination of these points,
y(t+h) + a2y(t) + a1y(th) = h{A2F(t) + A1F(th)}
where F(t) = f(t, y(t)). Our strategy is to choose the coefficients aiand Ai
to make the approximation consistent, meaning that the error is as small as
is reasonable.
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