Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations

Casas, Fernando; Chartier, Philippe; Escorihuela-Tomàs, Alejandro; Zhang, Yong

doi:10.1016/j.cam.2020.113006

Cited by 12 publications

(17 citation statements)

References 13 publications

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“…h have indeed close similarities with the compositions R (k) h (1.9) previously analyzed in [11]: not only their starting point is the same (the basic time-symmetric method S…”

Section: Methods T (K)supporting

confidence: 72%

“…that are still time-symmetric up to order 4n + 3. Moreover, if the differential equation in (1.1) has some qualitative properties (such as symplecticity or volume preservation) then these properties are still shared by the numerical solution up to order 4n + 3 [11].…”

Section: Discussionmentioning

confidence: 99%

“…as the associated series of operators, where each V j (h) is of the form (2.4). Now, by following the same approach as in [11], we express Φ(h) as…”

Section: Linear Combinations Of Symmetric-conjugate Compositionsmentioning

confidence: 99%

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High order integrators obtained by linear combinations of symmetric-conjugate compositions

Casas

Escorihuela-Tomàs

2022

Applied Mathematics and Computation

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A new family of methods involving complex coefficients for the numerical integration of differential equations is presented and analyzed. They are constructed as linear combinations of symmetric-conjugate compositions obtained from a basic time-symmetric integrator of order 2n (n ≥ 1). The new integrators are of order 2(n + k), k = 1, 2, . . ., and preserve time-symmetry up to order 4n + 3 when applied to differential equations with real vector fields. If in addition the system is Hamiltonian and the basic scheme is symplectic, then they also preserve symplecticity up to order 4n + 3. We show that these integrators are well suited for a parallel implementation, thus improving their efficiency. Methods up to order 10 based on a 4th-order integrator are built and tested in comparison with other standard procedures to increase the order of a basic scheme.

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“…h have indeed close similarities with the compositions R (k) h (1.9) previously analyzed in [11]: not only their starting point is the same (the basic time-symmetric method S…”

Section: Methods T (K)supporting

confidence: 72%

Section: Discussionmentioning

confidence: 99%

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High order integrators obtained by linear combinations of symmetric-conjugate compositions

Casas

Escorihuela-Tomàs

2022

Applied Mathematics and Computation

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“…Splitting and composition methods with complex coefficients, although computationally between 2 and 4 times more costly than their real counterparts when applied to ODEs involving real vector fields, possess however some remarkable properties: their truncation errors with the minimum number of stages are typically very small, and their stability threshold is comparatively large. Moreover, when the numerical solution is projected at each time step, they lead to approximations that still preserve important qualitative features (such as symplecticity and time-symmetry) up to an order much higher than the order of the method itself [7,9,4].…”

mentioning

confidence: 99%

“…It is said that the scheme is pseudo-symmetric of order 7, since it preserves the time-symmetry property up to terms of order h 7 [9].…”

mentioning

confidence: 99%

Applying splitting methods with complex coefficients to the numerical integration of unitary problems

Blanes

Casas

Escorihuela-Tomàs

2022

JCD

Self Cite

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<p style='text-indent:20px;'>We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schrödinger equation. We prove that a particular class of integrators are conjugate to unitary methods for sufficiently small step sizes when applied to problems defined in the group <inline-formula><tex-math id="M1">\begin{document}$ \mathrm{SU}(2) $\end{document}</tex-math></inline-formula>. In the general case, the error in both the energy and the norm of the numerical approximation provided by these methods does not possess a secular component over long time intervals, when combined with pseudo-spectral discretization techniques in space.</p>

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Hydrodynamics simulation of red blood cells: Employing a penalty method with double jump composition of lower order time integrator

Laadhari,

Deeb,

Kaoui

2023

Math Methods in App Sciences

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We propose a numerical framework tailored for simulating the dynamics of vesicles with inextensible membranes, which mimic red blood cells, immersed in a non‐Newtonian fluid environment. A penalty method is proposed to handle the inextensibility constraint by relaxation, allowing a simple computer implementation and an affordable computational load compared to the full mixed formulation. To handle the high‐order derivatives in the stress jump across the membrane, which arise due to the high geometric order of the Helfrich functional, we employ higher degree finite elements for spatial discretization. The time integration scheme relies on the double composition of the Crank–Nicolson scheme to achieve faster fourth‐order convergence behavior. Additionally, an adaptive time‐stepping strategy based on a third‐order temporal integration error estimation is implemented. We address the main features of the proposed method, which is benchmarked against existing numerical and experimental results. Furthermore, we investigate the influence of non‐Newtonian rheology on the system dynamics.

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Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations

Cited by 12 publications

References 13 publications

High order integrators obtained by linear combinations of symmetric-conjugate compositions

High order integrators obtained by linear combinations of symmetric-conjugate compositions

Applying splitting methods with complex coefficients to the numerical integration of unitary problems

Hydrodynamics simulation of red blood cells: Employing a penalty method with double jump composition of lower order time integrator

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