Euler’s method is both low-accuracy and usually unstable if it is used alone over many steps. But, it is $O(h)$ accurate and thus useful as an explicit estimate in $O(h)$ terms. An application of the trapezoidal rule yields $$\begin{aligned}
u(t+h) & = & u(t) + \frac{k}{2} \left( A u(t+h) + A u(t)
\right) \label{eq:trapezoidalrule} \\ & & \qquad + \frac{h}{2}
\left((1-u(t+h))u(t+h) + (1-u(t))u(t) \right) \nonumber\end{aligned}$$ and is an $O(h^{2}) + O((\Delta x)^{2})$ accurate procedure but solving the quadratic vector equation (\[eq:trapezoidalrule\]) for $u(t+h)$ is awkward. To the same $O(h^{2})$, we propose a semi-implicit procedure which uses the Euler estimate (\[eq:Eulerestimate\]) to replace one of the terms in (\[eq:trapezoidalrule\]): $$\begin{aligned}
u(t+h) & = & u(t) + \frac{k}{2} \left( A u(t+h) + A u(t) \right) \label{eq:semiimplicit}\\
& & \qquad + \frac{h}{2} \left( (1-u_{E})u(t+h) + (1-u(t))u(t) \right).\nonumber\end{aligned}$$ Equation (\[eq:semiimplicit\]) can now be solved as a linear system, $$\left( 1 - \frac{k}{2} A - \frac{h}{2} (1-u_{E}) \right) u(t+h) = 
u(t) + \frac{k}{2} A u(t) + \frac{h}{2} (1-u(t))u(t), 
\label{eq:semidifference}$$ because the matrix, $1 - \frac{k}{2} A - \frac{h}{2} (1-u_{E})$, on the left hand side is explicit, as is the right hand side. That is, this matrix and the right hand side contain only old data, 