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chapter 7 : time-dependent diāµerential equations
ordinary diāµerential equations
Let x(t) be the position of a particle moving on the x-axis at time t.
1st order ODE dx dt =^ f^ (x)^ :^ velocity is a function of position x(0) : initial position
The problem is to find the position x(t) for t > 0. ex
- dxdt = x , x(0) = 1 ) x(t) = e t
- dxdt = x 2 , x(0) = 1 ) x(t) = (^1) ^1 t
- dxdt = sin x , x(0) = 1 ) x(t) =? (^26) Thurs The simplest numerical method is Eulerās method.^4 /^18
choose t : time step
define w (^) n : numerical solution at time t (^) n = n t w (^) n+1 w (^) n t =^ f^ (w^ n^ )^ )^ w^ n+1^ =^ w^ n^ +^ tf^ (w^ n^ )
given w 0 , compute w 1 , w 2 ,...
questions : accuracy , stability , e ciency
2nd order ODE
d 2 x dt 2 =^ f^ (x)^ :^ acceleration is a function of position (Newtonās equation) x(0) , x 0 (0) : initial position , velocity w (^) n+1 2 w (^) n + w (^) n 1 ( t) 2 =^ f^ (w^ n^ )^ )^ w^ n+1^ = 2w^ n^ ^ w^ n ^1 + ( t)^
(^2) f (w (^) n )
given w 0 and w 1 , compute w 2 , w 3 ,...
partial diāµerential equations ex : heat equation u(x, t) : temperature of a metal rod at position x and time t @u @t =^ ļ£æ
@ 2 u @x 2 ,^ ļ£æ^ : coe cient of thermal expansion initial condition : u(x, 0) = f (x) , boundary conditions : u(0, t) = u(1, t) = 0 The simplest numerical method is a finite-diāµerence scheme. choose x : space step , t : time step define w nj : numerical solution at position x (^) j = j x and time t (^) n = n t w (^) jn +1 w (^) jn t =^ ļ£æ
w nj+1 2 w (^) jn + w (^) jn 1 ( x) 2 )^ w^
n j +1= w (^) jn + ļ£æ t ( x) 2 (w^ jn+1 ^2 w^ nj +^ w^ nj 1 ) case 1 : ļ£æ = 1 , x = 0. 05 , t = 0. 0013
(^00) 0.5 1
1 t=
(^00) 0.5 1
1 1 time step
(^00) 0.5 1
1 25 time steps
(^00) 0.5 1
1 50 time steps
case 2 : ļ£æ = 1 , x = 0. 05 , t = 0. 0012
(^00) 0.5 1
1 t=
(^00) 0.5 1
1 1 time step
(^00) 0.5 1
1 25 time steps
(^00) 0.5 1
1 50 time steps
explanation : the method is stable , (^) (ļ£æ xt) 2 ļ£æ (^12 ) Tues Lax equivalence theorem (Peter Lax)^4 /^23 Given a well-posed initial/boundary value problem and a consistent finite-diāµerence scheme, stability is necessary and su cient for convergence.
ex : wave equation
u(x, t) : displacement of a string at position x and time t
@ 2 u @t 2 =^ c^
2 @^2 u @x 2 c : wave speed
w (^) in +1 2 w (^) in + w ni ( t) 2 =^ c^
2 w^ in+1 ^2 w^ ni +^ w^ ni 1 ( x) 2
) w n i +1= 2w ni w n i ^1 + c xt
! 2 (w ni+1 2 w (^) in + w (^) in 1 ) , stable ,
c t x
ļ£æ^1
