B-convergence of the implicit midpoint rule and the trapezoidal rule

Abstract

Abstract.We present upper bounds for the global discretization error of the implicit midpoint rule and the trapezoidal rule for the case of arbitrary variable stepsizes. Specializing our results for the case of constant stepsizes they easily prove second order optimal B-convergence for both methods.1980

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

“…For stiff problems there may be damping or cancellation oflocal errors, as a result of which there can be convergence of order q while lbnl = O(hq) only. This was shown in [13] This convergence result shows that the order reduction is annihilated in the transition from local to global error. For stable one-leg schemes the order of convergence for stiff problems will be the same as in the nonstiff case.…”

“…If, on the other hand, the stepsize variation is restricted we consider the transformed and thus stability, and consequently convergence with order p, again follow from stability for the constant stepsize case, treated in Section 4. As for the one-leg methods, this can be proved for specific linear multistep methods under assumptions on the stepsize variation somewhat less restrictive than (5.2), but in contrast to the one-leg methods we may now have divergence for irregular grids, see for example the results of [13] for the trapezoidal rule.…”

“…A complete convergence analysis for the implicit midpoint rule and the trapezoidal rule applied to problems of the above type has been obtained by Kraaijevanger [13]. Our approach is closely related to this analysis.…”

“…This seems a reasonable assumption for numerical codes, where h. will be somehow related to the smoothness of solutions near t •. It was also shown in [13] that (5.8) is necessary to guarantee second order convergence; for stepsize sequences like h. = i;h (for n odd), h. = h (for n even) the order will reduce to 1. Thus we see that this method is also convergent of order 2, independently of the stiffness, if the grid refinement is such that (5.8) is satisfied.…”

On the error of general linear methods for stiff dissipative differential equations

“…In fact, it is known from Burrage & Hundsdorfer (1987) that there are only very few algebraically stable Runge-Kutta methods with property (4.5) and that the maximal classical order of such methods is 3. There seems to be only one Runge-Kutta method of practical interest for which we obtain convergence with order q + 1, namely the implicit midpoint rule, see Kraaijevanger (1985). The implicit midpoint rule can also be regarded as a one-leg multistep method.…”