Comparing Euler Polygons with Exact Solution Curves

Why Compare an Euler Polygon with an Exact Curve?

Euler’s method produces approximate values at selected times. Joining those values by straight line segments gives an Euler polygon, a continuous piecewise-linear approximation to the solution of an initial-value problem. When an exact solution is known, placing the polygon beside its solution curve lets us inspect how well the approximation represents the solution across an interval.

A fair comparison uses the same time for both values. At a grid point, compare the Euler value with the exact solution there. Between grid points, compare the point on the relevant line segment with the exact solution at that interior time. The sign of the difference says which graph is higher; its absolute value gives the size of the vertical discrepancy.

Definitions for a Point-by-Point Comparison

Consider an initial-value problem \(y'=f(t,y)\), \(y(t_0)=y_0\), and a grid \(t_0<t_1<\cdots<t_N\). Write \(h_n=t_{n+1}-t_n\) for the length of the \(n\)-th step. Forward Euler computes approximations \(y_n\) by

Suppose the exact solution \(y(t)\) is known on the interval being considered. The Euler polygon \(P(t)\) is the continuous function that has value \(y_n\) at \(t_n\) and is linear from \(y_n\) to \(y_{n+1}\) on each grid interval.

The distinction between signed and absolute discrepancy matters. For instance, \(E(t)=-0.2\) means the polygon is \(0.2\) units below the curve, while \(|E(t)|=0.2\) reports only the vertical distance. If the solution has physical units, these discrepancies have the same units as \(y\).

What the Polygon Does Between Grid Points

On the interval from \(t_n\) to \(t_{n+1}\), the polygon is the line joining \((t_n,y_n)\) and \((t_{n+1},y_{n+1})\). Its value is therefore

Here \(h_n>0\), and the formula works for a uniform or nonuniform grid. Substituting the Euler update gives an equivalent expression that makes the segment’s slope explicit:

This identity explains why the polygon segment has the slope used at the start of that Euler step. It also shows exactly how to find a polygon value at an interior time: use the starting approximation and the step’s slope, not a new Euler update at the interior time.

Worked Examples

A Common Mistake: Comparing Different Times

A discrepancy is meaningful only when the two compared values belong to the same time. For example, in the growth example, one might incorrectly compare the Euler value \(y_1=1.5\) at \(t=0.5\) with the exact value \(e^1\approx2.718282\) at \(t=1\). That gives \(1.5-2.718282\approx-1.218282\), but it is not an error at either time: it subtracts values from different times.

The correction is to evaluate the exact solution at \(t=0.5\): \(e^{0.5}\approx1.648721\). The matching-time discrepancy is \(1.5-1.648721\approx-0.148721\). A related mistake is to call a negative signed discrepancy a negative distance. Distance is nonnegative; the sign instead records which graph is higher.

What the Comparison Can—and Cannot—Show

Plotting both graphs and calculating discrepancies at selected times helps describe the approximation on a fixed interval. The signed values indicate whether the polygon is above or below the curve at those times; absolute values allow comparisons of vertical separation. Interior evaluations are useful because matching grid-point values alone do not describe every point along the straight segments.

These comparisons describe what happens for the chosen equation, initial value, grid, and interval. They do not by themselves prove that the polygon stays on one side of the curve between sampled times, nor that the discrepancy changes monotonically throughout the interval. A visual impression is useful, but it is not a general error bound. The later analysis of Euler error addresses questions of local and accumulated error more systematically.