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The generalized trapezoid algorithm provides access to three
specific timestepping methods depending on the value of
:
forward Euler (
), trapezoid rule (
), and backward Euler (
).
- Any method with
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. - The trapezoid rule (
) is second order
accurate in time. All other values of
produce first order
methods. - Backward Euler (
) 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.
Dan Yergeau
Wed Jun 18 19:17:04 PDT 1997