Symmetrized local error estimators for time-reversible one-step methods in nonlinear evolution equations

Abstract

Prior work on computable defect-based local error estimators for (linear) timereversible integrators is extended to nonlinear and nonautonomous evolution equations. We prove that the asymptotic results from the linear case [W. Auzinger and O. Koch, An improved local error estimator for symmetric time-stepping schemes, Appl. Math. Lett. 82 (2018), pp. 106-110] remain valid, i.e., the modified estimators yield an improved asymptotic order as the step size goes to zero. Typically, the computational effort is only slightly higher than for conventional defect-based estimators, and it may even be lower in some cases. We illustrate this by some examples and present numerical results for evolution equations of Schrödinger type, solved by either time-splitting or Magnus-type integrators. Finally, we demonstrate that adaptive time-stepping schemes can be successfully based on our local error estimators.

Optimal parameters for numerical solvers of PDEs

“…The figures in Table 5. 4 show that with increasing r the computation time decreases while the accuracy is not significantly affected even for r = 10. This is a consequence of the fact that the sequence of parameters obtained for different values of r quickly approach each other after just a few time-steps (see Figure 5.9).…”

“…quantifies the extent to which the numerical flow Φ fails to satisfy (2.2). For nonlinear parabolic PDEs and time-reversible equations, the following nonlinear variationof-constant formula holds true [4,11]. Henceforth, ∂ k f denotes the Fréchet derivative of a function f with respect to the k-th argument.…”

“…In the context of backward error analysis, the defect measures the discrepancy between the differential equation satisfied by the numerical solution and the original equation [26]. Defect based error estimates have been utilized widely in the development of time-adaptive methods for ordinary differential equations (ODEs) [12,21] and PDEs [3,4,6], but, to the best of our knowledge, these have not been employed for the estimation of optimal parameters. Unlike adaptive techniques for choosing time-steps, where the local error can be assumed to decrease monotonically with the time-step, in the proposed approach an optimization problem needs to be solved for finding the values of the parameters.…”

Efficient Magnus-type integrators for solar energy conversion in Hubbard models

“…When we consider self-adjoint (or symmetric) schemes which are characterized by the identity S(−τ ; t 0 + τ ) S(τ ; t 0 ) = Id, (3.12) a higher asymptotical quality of the error estimator can be obtained at moderate additional expense by introducing a symmetrized version of the defect, which was introduced and analyzed in [33,34]. We define…”

Time adaptive Zassenhaus splittings for the Schrödinger equation in the semiclassical regime

“…An alternative way of defining the defect was introduced in [17,18]. It is based on the fact that the exact evolution operator E(t) commutes with the Hamiltionian H, whence…”