Symmetric-conjugate splitting methods for evolution equations of parabolic type

TL;DR

This work analyzes higher-order symmetric-conjugate operator splitting methods for linear parabolic problems and compares them to standard symmetric splits. By allowing complex coefficients with nonnegative real parts, these methods can achieve high-order stability while preserving a self-adjoint evolution operator, which keeps imaginary-part errors bounded and energies well-behaved. A key insight is that local error estimates can be efficiently obtained from the imaginary parts, enabling adaptive time stepping without extra costs. Numerical experiments on parabolic and Schrödinger-type problems confirm the theoretical advantages, including robust ground-state computations and effective adaptive schemes, highlighting the practical potential for nonreversible systems and imaginary-time propagation.

Abstract

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.

Figures (10)

• Figure 1: Real and complex splitting methods applied in numerical tests. Denominations and characteristics (nonstiff order $p$, number of stages $s$). The coefficients of the symmetric-conjugate schemes are given in Figures \ref{['fig:Figure2']} and \ref{['fig:Figure3']}.
• Figure 2: Time integration of the parabolic model problem \ref{['eq:TestProblem1']} by non-optimised and optimised fourth-order operator splitting methods involving complex coefficients with time increments $h = \frac{T}{40}$ (top curves) and $h=\frac{T}{400}$ (bottom curves). Relative errors in the imaginary parts of the numerical solutions over time (left) and corresponding errors in the ground state energy (right). Symmetric schemes comprising $s = 4$ (thin black dashed line) and $s > 4$ (thin black solid line) stages with increasing errors in a log-log scale versus symmetric-conjugate schemes comprising $s = 4$ (thick red dashed line) and $s > 4$ (thick red solid line) stages with bounded errors.
• Figure 3: Time integration of the linear parabolic model problem with real-valued solution by real and complex splitting methods, see also \ref{['eq:ModelProblems']} and Figure \ref{['fig:Figure1']}. For the considered quadratic potential, the exact solution is known. Left: Local and global errors. Right: Corresponding errors in the imaginary parts.
• Figure 4: Time integration of the linear parabolic model problem with real-valued solution by real and complex splitting methods, see also \ref{['eq:ModelProblems']} and Figure \ref{['fig:Figure1']}. For the considered quartic potential, a numerical reference solution is computed. Left: Local and global errors. Right: Corresponding errors in the imaginary parts.
• Figure 5: Adaptive time integration of a linear parabolic model problem with known real-valued solution for $t_0 = 0$ and $T = 1$ by a third-order symmetric-conjugate splitting method, see also \ref{['eq:ModelProblems']} and Figure \ref{['fig:Figure1']}. The local error estimation is based on the computation of the imaginary parts with respect to the Euclidean norm and requires negligible additional costs. The total numbers of time steps 47 and 997 are adjusted in accordance with the prescribed tolerances $10^{-6}$ and $10^{-10}$, respectively. Left: Sequences of time grid points. Right: Associated sequences of local errors determined with respect to the exact solution values.