The generalized trapezoid algorithm provides access to three
specific timestepping methods depending on the value of 
:
forward Euler ( 
), trapezoid rule ( 
), and backward Euler ( 
).
is conditionally
A-stable; time-steps are constrained by the maximum eigenvalue of the
discretized system. Arbitrarily large time-steps will produce an
amplification matrix with entries larger than 1 in magnitude. This
can lead to incorrect exponential growth in the solution. ) is second order
accurate in time. All other values of produce first order
methods. ) is the only method that is both
A-stable and L-stable. With , the entries in the
amplification matrix are bounded by [0,1] (L-stable). Other
values of ( 
) can produce
amplification matrices with values near -1 leading to
solutions with oscillatory behavior.