Symmetric-conjugate splitting methods for evolution equations of parabolic type

S. Blanes; F. Casas; C. González; M. Thalhammer

doi:10.3934/jcd.2024003

Article Contents

2024, Volume 11, Issue 1: 108-134. Doi: 10.3934/jcd.2024003

This issue Previous Article B-stability of numerical integrators on Riemannian manifolds Next Article Preface

Symmetric-conjugate splitting methods for evolution equations of parabolic type

1.
Universitat Politècnica de València, Instituto de Matemática Multidisciplinar, 46022 Valencia, Spain
2.
Universitat Jaume I, IMAC and Departament de Matemàtiques, 12071 Castellón, Spain
3.
Universidad de Valladolid, Departamento de Matemática Aplicada, 47011 Valladolid, Spain
4.
Universität Innsbruck, Institut für Mathematik, 6020 Innsbruck, Austria

^*Corresponding author: Mechthild Thalhammer
^*Corresponding author: Mechthild Thalhammer

Received: August 2023

Revised: December 2023

Early access: January 22, 2024

Published online: January 22, 2024

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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.

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Mathematics Subject Classification: 65J10, 65L04, 65M12.

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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 9 and 10

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Figure 2. Time integration of the parabolic model problem (20) 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

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Figure 3. Time integration of the linear parabolic model problem with real-valued solution by real and complex splitting methods, see also (8) and Figure 1. For the considered quadratic potential, the exact solution is known. Left: Local and global errors. Right: Corresponding errors in the imaginary parts

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Figure 4. Time integration of the linear parabolic model problem with real-valued solution by real and complex splitting methods, see also (8) and Figure 1. For the considered quartic potential, a numerical reference solution is computed. Left: Local and global errors. Right: Corresponding errors in the imaginary parts

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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 (8) and Figure 1. 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

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Figure 6. Corresponding results for a sixth-order symmetric-conjugate splitting method and lower tolerances $ 10^{-10} $ and $ 10^{-12} $, respectively

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Figure 7. Corresponding results for the third-order symmetric-conjugate splitting method with respect to the maximum norm

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Figure 8. Corresponding results for the sixth-order symmetric-conjugate splitting method with respect to the maximum norm

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Figure 9. Coefficients of symmetric-conjugate operator splitting methods applied in numerical tests

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Figure 10. Coefficients of symmetric-conjugate operator splitting methods applied in numerical tests

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Table 1. Stability properties of real and complex splitting methods in the context of parabolic equations. Schemes with non-negative coefficients $ (a_j)_{j = 1}^{s} $ remain stable for Schrödinger equations

Lie–Trotter (real, $ p = 1 $, $ s = 1 $)	Stability ($ a_1> 0 $)
Strang (real, $ p = 2 $, $ s = 2 $)	Stability ($ a_1, a_2 \geq 0 $)
Yoshida (real, $ p = 4 $, $ s = 4 $)	Instability ($ a_3< 0 $)
Complex (symmetric, $ p = 4 $, $ s = 4 $)	Stability ($ \Re \, a_1, \dots, \Re \, a_s \geq 0 $)
Complex (symmetric, $ p = 4 $, $ s = 5 $)	Stability ($ a_1, \dots, a_s \geq 0 $)
Complex (symmetric, $ p = 4 $, $ s = 6 $)	Stability ($ a_1, \dots, a_s \geq 0 $)
Complex (symmetric, $ p = 6 $, $ s = 17 $)	Stability ($ a_1, \dots, a_s \geq 0 $)
Complex (symmetric-conj., $ p = 3 $, $ s = 3 $)	Stability ($ \Re \, a_1, \dots, \Re \, a_s \geq 0 $)
Complex (symmetric-conj., $ p = 3 $, $ s = 4 $)	Stability ($ a_1, \dots, a_s \geq 0 $)
Complex (symmetric-conj., $ p = 4 $, $ s = 4 $)	Stability ($ \Re \, a_1, \dots, \Re \, a_s \geq 0 $)
Complex (symmetric-conj., $ p = 4 $, $ s = 6 $)	Stability ($ a_1, \dots, a_s \geq 0 $)
Complex (symmetric-conj., $ p = 6 $, $ s = 12 $)	Stability ($ a_1, \dots, a_s \geq 0 $)
Complex (symmetric-conj., $ p = 6 $, $ s = 16 $)	Stability ($ a_1, \dots, a_s \geq 0 $)

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Table 2. Application of real and complex splitting methods to parabolic model problems with real-valued solutions. List of classical orders (first column), numerically observed orders of convergence for solution values (second column), and numerically observed orders for imaginary parts (third column)

Lie–Trotter (real, $ p = 1 $)	$ p_{\, \text{num}} = p = 1 $	——
Strang (real, $ p = 2 $)	$ p_{\, \text{num}} = p = 2 $	——
Yoshida (real, $ p = 4 $)	$ p_{\, \text{num}} = p = 4 $	——
Complex (symmetric, $ p = 4 $)	$ p_{\, \text{num}} = p = 4 $	$ p_{\, \text{num}, \Im} = p = 4 $
Complex (symmetric, $ p = 6 $)	$ p_{\, \text{num}} = p = 6 $	$ p_{\, \text{num}, \Im} = p = 6 $
Complex (symmetric-conj., $ p = 3 $)	$ p_{\, \text{num}} = p = 3 $	$ p_{\, \text{num}, \Im} = p = 3 $
Complex (symmetric-conj., $ p = 4 $)	$ p_{\, \text{num}} = p = 4 $	$ p_{\, \text{num}, \Im} = p + 1 = 5 $
Complex (symmetric-conj., $ p = 6 $)	$ p_{\, \text{num}} = p = 6 $	$ p_{\, \text{num}, \Im} = p + 1 = 7 $

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