Introduction to scientific Computing A Matrix Vector Approach Using Matlab Written by Charles FVan Loan 陈文斌 Wbchen(fudan. edu. cn 复日大学
Introduction to Scientific Computing -- A Matrix Vector Approach Using Matlab Written by Charles F.Van Loan 陈 文 斌 Wbchen@fudan.edu.cn 复旦大学
Chapter 9 The Initial value Problem Ba asic concepts The Runge-Kutta Methods The adams methods
Chapter 9 The Initial Value Problem • Basic concepts • The Runge-Kutta Methods • The Adams Methods
()=-5 Solutions to y()=-5 y(t) 0.6 0.2 0.150.2 0.35
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Solutions to y'(t) = -5 y(t) y'(t) = −5t
Euler method Five Steps of Euler Method (y'=-5y, y(o)=1) n+1 n+h,f(tn,yu) 16 0.8 9105 0.0 0.1 0.15 0.2 0.25 0.35
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Five Steps of Euler Method (y'=-5y, y(0)=1) ( , ) n 1 n n n n y = y + h f t y + Euler Method
Theorem 9 Assume that, for n=0: N, a function y, (t) exists that solves the ivp y(t=f(t, y(t), y(t,=y where(to,yo),…,(tN,y) are given and t<t1<…< Define the global error by gn=yo(tn)-y and the local truncatio n error by LTEM=Ym(tn)-y If for all tElto, tN]and none of the trajector ies {(t,yn():≤t≤N} =0 intersect. then for n=1: N gn≤∑|LTEk k=1
= − = = = = − = − = = = 1 0 0 N 1 0 0 0 N 0 1 | | | | intersect, then for 1: {( , ( )) : } 0 : for all [ , t ] and none of the trajectories 0 ( , ) If ( ) . and the local truncatio n error by Define the global error by ( ) where ( , y ),...,( , y ) are given and . ( )), ( ) solves the IVP Theorem 9 Assume that, for 0 : , a function (t) exists that k n k n y n n n n n n n N N n n n g LTE n N t y t t t N n n t t y f t y f LTE y t y g y t y t t t t t y'(t) f(t,y t y t y n N y