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Summary

DescriptionODE Exp Imp Euler method.svg	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'=-200z, 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	4 July 2010
Source	Own work
Author	LutzL
SVG development InfoField	The SVG code is valid. This plot was created with gnuplot. Category:Valid SVG created with Gnuplot#ODE%20Exp%20Imp%20Euler%20method.svg Category:Translation possible - SVGThis plot uses embedded text that can be easily translated using a text editor.

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Source

Using a small C program and gnuplot.

exp_imp_euler.c

/*
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

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

Category:Euler method Category:Images with C source code Category:Images with Gnuplot source code Category:Numerical analysis