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  3. 线性多步(Adams)和预估校正法(改进的欧拉法)的比较

线性多步(Adams)和预估校正法(改进的欧拉法)的比较

mac2026-08-03  1

线性多步和预估校正法的比较

问题描述两步的隐式Adams公式改进的欧拉公式误差阶MATLAB代码

问题描述

求解常微分方程: { d y d x = x 3 − y x , y ( 1 ) = 2 5 . \begin{cases} \dfrac{dy}{dx}=x^3-\frac{y}{x},\\ y(1)=\frac{2}{5}. \end{cases} ⎩⎨⎧​dxdy​=x3−xy​,y(1)=52​.​ 该方程的精确解为: y = 1 5 x 4 + 1 5 x . y=\frac{1}{5}x^4+\frac{1}{5x}. y=51​x4+5x1​.

两步的隐式Adams公式

y n + 1 = y n + h 2 ( f ( x n + 1 , y n + 1 ) + f ( x n , y n ) ) . y_{n+1}=y_n+\frac{h}{2}(f(x_{n+1},y_{n+1})+f(x_n,y_n)). yn+1​=yn​+2h​(f(xn+1​,yn+1​)+f(xn​,yn​)). 对于这个问题可以被写为: ( 1 + h 2 x n + 1 ) y n + 1 = ( 1 − h 2 x n ) y n + h 2 ( x n + 1 3 + x n 3 ) . (1+\frac{h}{2x_{n+1}})y_{n+1}=(1-\frac{h}{2x_n})y_n+\frac{h}{2}(x_{n+1}^3+x_n^3). (1+2xn+1​h​)yn+1​=(1−2xn​h​)yn​+2h​(xn+13​+xn3​).

改进的欧拉公式

{ y n + 1 ∗ = y n + h f ( x n , y n ) , y ( n + 1 ) = y n + h 2 ( f ( x n , y n ) + f ( x n + 1 , y n + 1 ∗ ) ) . \begin{cases} y_{n+1}^{*}=y_n+hf(x_n,y_n),\\ y(n+1)=y_n+\frac{h}{2}(f(x_n,y_n)+f(x_{n+1},y_{n+1}^{*})). \end{cases} {yn+1∗​=yn​+hf(xn​,yn​),y(n+1)=yn​+2h​(f(xn​,yn​)+f(xn+1​,yn+1∗​)).​

误差阶

误差阶( l ∞ l_\infty l∞​)Adams改进的欧拉法h=0.12.0002741.999845h=0.052.0000681.999961h=0.0012.0000031.999998h=0.00052.0000012.000000

MATLAB代码

%建立error_order.m文件 clear all,close all,clc [error1,error2]=shili(0.005); e1=[norm(error1,inf),norm(error2,inf)]; [error1,error2]=shili(0.0025); e2=[norm(error1,inf),norm(error2,inf)]; format long error_order_linf=log2(e1./e2) %建立shili.m文件 %clear all,close all,clc function [error1,error2]=shili(h) f=@(x,y)x^3-y/x; %h=0.1; x=[1:h:2]; N=size(x,2)-1; %% Adams y1=[2/5,zeros(1,N)]; for n=1:N mid=(1-h/(2*x(n)))*y1(n)+h/2*(x(n+1)^3+x(n)^3); y1(n+1)=mid/(1+h/(2*x(n+1))); end %% Improved Euler method y2=[2/5,zeros(1,N)]; for n=1:N y2(n+1)=y2(n)+h*f(x(n),y2(n)); y2(n+1)=y2(n)+h/2*(f(x(n),y2(n))+f(x(n+1),y2(n+1))); end y=1/5*x.^4+1./(5*x); % Exact solution error1=y-y1; error2=y-y2; end
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