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The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant applications include nonreversible systems and ground state computations for linear Schrödinger equations based on the imaginary time propagation. Numerical examples confirm the favourable error behaviour of higher-order symmetric-conjugate splitting methods and illustrate the usefulness of a time stepsize control, where the local error estimation relies on the computation of the imaginary parts and thus requires negligible costs. [ABSTRACT FROM AUTHOR]

In the present paper, stability and convergence properties of linear multistep methods are investigated. The attention is focused on parabolic problems and variable stepsizes. Under weak assumptions on the method and the stepsize sequence an asymptotic stability result is shown. Further, stability bounds for linear nonautonomous parabolic problems with Ho¨lder continuous operator are given. With the help of these results, convergence estimates for semilinear and fully nonlinear parabolic problems are derived. [Copyright &y& Elsevier]

Abstract: We analyse stability and convergence properties of a second-order Magnus-type integrator for linear parabolic differential equations with time-dependent coefficients, working in an analytic framework of sectorial operators in Banach spaces. Under reasonable smoothness assumptions on the data and the exact solution, we prove a second-order convergence result without unnatural restrictions on the time stepsize. However, if the error is measured in the domain of the differential operator, an order reduction occurs, in general. A numerical example illustrates and confirms our theoretical results. [Copyright &y& Elsevier]