sec 21: generalizations of the euler method consider a differential equation n = 10 estimate x = 0.5...
DESCRIPTION
Sec 21: Numerical Integration Left Riemann sum Right Riemann sum Explicit Euler Method Implicit Euler MethodTRANSCRIPT
Sec 21: Generalizations of the Euler Method
Consider a differential equation
n = 10 estimate x = 0.5n = 10 estimate x =50
Initial Value ProblemEuler Method
suffers from some limitations.
the error behaves only like thefirst power of h.
In the present section, we consider generalizations, which will yield improved numerical behaviour but will retain, as much as possible, its characteristic property of simplicity.
Sec 21: Numerical Integration
Left Riemann sum
)()(1
k
x
xxhfdxxfk
k
kx 1kx
Right Riemann sum
)()( 11
k
x
xxhfdxxfk
k
kx 1kx
Midpoint
)()( 21
1
kkk
k
xxx
xhfdxxf
kx 1kx
Trapezoidal rule
)()(2
1 )( kk
k
k
xfxfhx
xdxxf
kx 1kx
Sec 21: Numerical Integration
Left Riemann sum
)()(1
k
x
xxhfdxxfk
k
kx 1kx
Right Riemann sum
)()( 11
k
x
xxhfdxxfk
k
kx 1kx
Explicit Euler Method
))(,()(' xyxfxy
11 ))(,()(' k
k
k
k
x
x
x
xxyxfdxxy
Implicit Euler Method
))(,()(' xyxfxy
11 ))(,()(' k
k
k
k
x
x
x
xxyxfdxxy
)~,(~~1 kkkk yxhfyy
)~,(~~111 kkkk yxhfyy
Sec 221: More computations in a step
Trapezoidal rule
)()(2
1 )( kk
k
k
xfxfhx
xdxxf
kx 1kx 11 ))(,()(' k
k
k
k
x
x
x
xxyxfdxxy
This is an example of a Runge–Kutta method
)~,( 12 KyhxhfK kk
)~,( 111 kk yxhfK
),(~~212
11 KKyy kk
Heun’s method
The interpolation polynomial
Approximate the function f(x) by a polynomial
)()()(1
1
kk
kk xx
xxxl
Lagrange Polynomial
kx 1kx1kx
)(xf
)()()(
11
kk
kk xx
xxxl
ijji xl )(
),( kk yx
),( 11 kk yx
),( 11 kk yx
polynomialxf )(
interpolating polynomial
)(),()(),()()( 111 xlyxfxlyxfxPxf kkkkkk
222 Greater dependence on previous values (linear multistep method)
)()()(1
1
kk
kk xx
xxxl
Lagrange Polynomial
kx 1kx1kx
)(xf
)()()(
11
kk
kk xx
xxxl
ijji xl )( ),( kk yx
),( 11 kk yx
),( 11 kk yx
two-step Adams-bashforth method
)(),()(),()()( 111 xlyxfxlyxfxPxf kkkkkk
11 ))(,()(' k
k
k
k
x
x
x
xxyxfdxxy
1 )(),()(),( 111k
k
x
x kkkkkk dxxlyxfxlyxf
)~,()~,(~~112
123
1 kkkkkk yxhfyxhfyy
223 Use of higher derivatives (Taylor series method)
Consider
By differentiating
Taylor series
If these higher derivatives are available, then the most popular option is to use them to evaluate a number of terms in Taylor’s theorem.
224 Multistep–multistage–multiderivative methods
While multistep methods, multistage methods and multiderivative methods all exist in their own right, many attempts have been made to combine their attributes so as to obtain new methods of greater power.