File:ODE Exp Imp Euler method.svg
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Summary[edit]
| Description |
English: Compares the forward/explicit and backward/implicit Euler methods for the numerical solution of the ODE y'(x)=-200*(y-cos(x))-sin(x), y(0)=1.
With the substituton z=y-cos(x), the ODE z'=-200*z, z(0)=0, results. Numerical integration of the second ODE should always result in the constant 0. By the nonlinear substitution in the first ODE, numerical errors are introduced in every step that are magnified by the factor 200. For not too small stepsizes, this results in explosive behavior of the explicit method whereas the implicit method remains close to the exact solution. |
| Date | |
| Source | Own work |
| Author | LutzL |
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. |
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Source[edit]
Using a small C program and gnuplot.
exp_imp_euler.c[edit]
/*
implizites und explizites Eulerverfahren für y'(x)=-2000*(y-cos(x))-sin(x)
*/
#include<stdio.h>
#include<math.h>
#define ALPHA 200
double f(double x,double y) {
return -ALPHA*(y-cos(x))-sin(x);
}
double explicit(double x,double y,double h) {
return y+h*f(x,y);
}
double implicit(double x,double y, double h) {
/* solve yp=y+h*f(xp,yp) for yp
explicit solution is
y=yp-h*f(xp,yp)
=(1+2000*h)*yp-2000*h*cos(xp)+h*sin(xp)
*/
double xp=x+h;
return (y+h*(ALPHA*cos(xp)-sin(xp)))/(1+ALPHA*h);
}
int main(){
double x,ye,yi,h;
FILE *fp=fopen("imp_euler.dat","w");
x=0; ye=yi=1; h=1.2e-2;
for(;x<1; x+=h){
fprintf(fp,"%8.5f\t%8.5f\t%8.5f\t%8.5f\n",x,cos(x),ye,yi);
ye=explicit(x,ye,h);
yi=implicit(x,yi,h);
}
return 0;
}
gnuplot commands[edit]
set term push
set term svg size 1200,900 fsize 28 linewidth 2
set output "exp_imp_euler.svg"
pl [:] [0:2] 'imp_euler.dat' u 1:3 ti 'explizit' w po pt 3, 'imp_euler.dat' u 1:4 w lin lt 3 lw 3 ti 'implizit'
unset outp
set term pop
File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 16:30, 4 July 2010 | | 1,200 × 900 (10 KB) | LutzL (talk | contribs) | {{Information |Description={{en|1=Compares the forward/explicit and backward/implicit Euler methods for the numerical solution of the ODE y'(x)=-200*(y-cos(x))-sin(x), y(0)=1. With the substituton z=y-cos(x), the ODE z'=-200*z, z(0)=0, results. Numerica |
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