Positivity and Equilibrium Preservation in Euler Steps

Why Preservation Properties Matter

Numerical solutions are often expected to respect features of the differential equation being approximated. If a system has a stationary state, an Euler computation should ideally leave that state stationary. If a quantity represents a population, concentration, or amount, negative numerical estimates may be meaningless even when the differential equation's solution remains nonnegative.

Forward Euler does not preserve every desirable feature automatically. The update uses the slope at the beginning of a step and extends that slope across the full step. A sufficiently large step can therefore move an estimate past zero. Understanding the update itself gives precise tests for when equilibrium and nonnegativity are preserved.

Forward Euler and Equilibria

For an initial-value problem \(y'=f(t,y)\), forward Euler with step size \(h>0\) uses \(t_{n+1}=t_n+h\) and the update \(y_{n+1}=y_n+h f(t_n,y_n)\). For an autonomous equation \(y'=f(y)\), an equilibrium is a value \(y_*\) satisfying \(f(y_*)=0\). At an equilibrium the differential equation has zero slope, so the constant function \(y(t)=y_*\) is a solution.

The reason is direct: substituting the equilibrium into the update gives \(y_{n+1}=y_*+h\cdot 0=y_*\). Applying the same calculation at every later step shows that the computed values remain at \(y_*\). This property is exact; it does not depend on the size of the step.

For a nonautonomous equation \(y'=f(t,y)\), a constant state \(y_*\) is an equilibrium solution on an interval only if \(f(t,y_*)=0\) for every time in that interval. If the slope is zero only at the current point \((t_n,y_n)\), then that particular Euler update leaves the value unchanged, but the slope may be nonzero at the next time. One zero-slope step does not by itself establish an equilibrium solution.

What Positivity Preservation Requires

Suppose a scalar quantity begins nonnegative, so \(y_n\geq 0\). The next Euler value is nonnegative exactly when

This is the exact one-step test at the state being used. If \(y_n>0\), it can also be written \(f(t_n,y_n)\geq -y_n/h\). In other words, the downward slope must not be so steep that its Euler line crosses zero during the step. When \(y_n=0\), the test reduces to \(f(t_n,0)\geq 0\).

A useful uniform sufficient condition is to bound how negative the slope can be relative to the state. Suppose \(c\geq 0\), and throughout the nonnegative states of interest the equation satisfies \(f(t,y)\geq -cy\). Then an Euler step with \(hc\leq 1\) cannot produce a negative value. This condition can be checked before carrying out the steps and is often easier to use than checking each update separately.

Why the Step-Size Bound Works

The proof is an estimate using the assumed lower bound on the slope. Because \(h>0\), multiplying \(f(t_n,y_n)\geq -cy_n\) by \(h\) preserves the inequality. Adding \(y_n\) to both sides gives

The step-size condition \(hc\leq 1\) implies \(1-hc\geq 0\). Since \(y_n\geq 0\), the product \((1-hc)y_n\) is nonnegative, and therefore \(y_{n+1}\geq 0\). Repeating this argument at each step proves the final claim.

The bound is sufficient, not always necessary. A particular update may remain nonnegative even when \(hc>1\), because its actual slope may be less negative than the lower bound allows. Conversely, the condition \(f(t,0)\geq 0\) alone is not enough: it checks the slope at the boundary, not whether a step from a positive state can jump across that boundary.

Decay: Positivity and Step Size

For the decay equation \(y'=-ky\), where \(k>0\), the Euler update is \(y_{n+1}=(1-kh)y_n\). Nonnegative input values stay nonnegative when \(1-kh\geq 0\), which requires \(h\leq 1/k\). This is the positivity bound for this equation. It should not be confused with a broader stability condition: a computation can avoid negative values yet still have other numerical behavior that needs checking.

Boundary Slopes Are Not Enough

Consider \(y'=4-y\). At the boundary \(y=0\), the slope is \(4\), pointing toward positive values. But for a sufficiently large positive \(y\), the slope is negative, and a large Euler step can cross below zero. In fact, the Euler update is \(y_{n+1}=(1-h)y_n+4h\). For all \(y_n\geq 0\), the inequality \(4-y_n\geq -y_n\) holds, so the theorem applies with \(c=1\) whenever \(h\leq 1\).

Positivity in a Coupled Linear System

The same reasoning applies component by component to a system. For a vector \(\mathbf{y}\), nonnegative means that every component is nonnegative. For the linear system \(\mathbf{y}'=\mathbf{A}\mathbf{y}+\mathbf{b}\), Euler gives \(\mathbf{y}_{n+1}=(\mathbf{I}+h\mathbf{A})\mathbf{y}_n+h\mathbf{b}\), where \(\mathbf{I}\) is the identity matrix. A sufficient condition for preserving nonnegative vectors is that every entry of \(\mathbf{I}+h\mathbf{A}\) and \(h\mathbf{b}\) is nonnegative. For example, if the off-diagonal entries of \(\mathbf{A}\) and the entries of \(\mathbf{b}\) are nonnegative, it remains to require \(1+hA_{ii}\geq 0\) for each diagonal entry.

Common Mistakes and Practical Checks

A frequent error is to reason that because the differential equation points into the nonnegative region at zero, Euler must remain nonnegative. That conclusion overlooks the size of a step from states away from zero. The affine example showed a positive slope at zero alongside a negative Euler result for a large step.

Another mistake is to identify positivity with equilibrium preservation. Equilibrium preservation follows from substituting a zero slope into the update and holds for any positive step size. Positivity requires a separate inequality involving the step size and the negative slopes that may occur. For \(y'=-3y\), for instance, the equilibrium \(y=0\) is preserved for every \(h\), while nonnegative values are guaranteed to remain nonnegative only when \(h\leq 1/3\).

Before computing a sequence of Euler values for a model that should remain nonnegative:

• Identify the state region that is physically or mathematically relevant.
• Write the Euler update and test whether its value is nonnegative at the boundary and at positive states.
• Look for a bound such as \(f(t,y)\geq -cy\), then impose \(hc\leq 1\).
• For a coupled system, check the signs of the Euler matrix and any added vector.
• Verify the resulting update directly for representative states, without treating those checks as a replacement for a general condition.

These tests concern preservation of a property, not the overall accuracy of the approximation. A nonnegative output need not be accurate, and preserving an equilibrium does not ensure that nearby numerical values behave like nearby solutions.