Performance of Borel–Padé–Laplace integrator for the solution of stiff and non-stiff problems

Deeb, Ahmad; Hamdouni, Aziz; Razafindralandy, Dina

doi:10.1016/j.amc.2022.127118

Cited by 7 publications

(8 citation statements)

References 35 publications

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“…However, in the context of stiff problems, the RK4 scheme demonstrates a slower improvement of accuracy, posing difficulties in attaining high levels of precision. This observation aligns with some remarks reported in the literature, see, for example, Deeb et al [52].…”

Section: Figure 4 Examplesupporting

confidence: 93%

“…The step size will be based on an error estimate of the temporal approximation, set up to a predefined accuracy requirement. Higher order numerical approximations can be obtained using integrators based on divergent series resummation [50], which have been applied to large time dynamics problems [51], stiff problems [52], and diffusion equations within the framework of proper generalized decomposition [53].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Figure 4 Examplesupporting

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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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show abstract

“…Time integration. In numerical time integration for dynamical systems, it is very important to be careful with the integration part when it comes to stiff equations (Deeb et al, 2022;Ernst and Gerhard, 1996), and especially when it comes to large time dynamics (Razafindralandy et al, 2019), as in the model of heat and mass transfer. Numerical schemes were massively developed to simulate the time evolution of the solution.…”

Section: Numerical Modelmentioning

confidence: 99%

Development of a hysteresis model based on axisymmetric and homotopic properties to predict moisture transfer in building materials

Deeb

Benmahiddine

Berger

et al. 2023

Journal of Building Physics

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Current hygrothermal behaviour prediction models neglect the hysteresis phenomenon. This leads to a discrepancy between numerical and experimental results, and a miscalculation of buildings’ durability. In this paper, a new mathematical model of hysteresis is proposed and implemented in a hygrothermal model to reduce this discrepancy. The model is based on a symmetry property between sorption curves and uses also a homotopic transformation relative to a parameter [Formula: see text]. The advantage of this model lies in its ease of use and implementation since it could be applied with the knowledge of only one main sorption curve by considering [Formula: see text], in other words, we only use the axisymmetric property here. In the case where the other main sorption curve is known, we use this curve to incorporate the homotopy property in order to calibrate the parameter [Formula: see text].The full version of the proposed model is called Axisymmetric + Homotopic. Furthermore, it was compared not only with the experimental sorption curves of different types of materials but also with a model that is well known in the literature (CARMELIET’s model). This comparison shows that the Axisymmetric + Homotopic model reliably predicts hysteresis loops of various types of materials even with the knowledge of only one of the main sorption curves. However, the full version of Axisymmetric + Homotopic model is more reliable and covers a large range of materials. The proposed model was incorporated into the mass transfer model. The simulation results strongly match the experimental ones.

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“…Then, it was extended by Deeb et al [14] to solve PDEs in the FEM Framework (FEMF). Other types of problems were also considered by this algorithm as stiff ones [15] and problems with large time dynamics [38].…”

Section: Introductionmentioning

confidence: 99%

Stabilized time‐series expansions for high‐order finite element solutions of partial differential equations

Deeb,

Dutykh

2024

Stud Appl Math

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Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time‐Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the application of high‐order Finite Element (FE) to certain classes of PDEs, such as diffusion equations and the Navier–Stokes (NS) equations, often leads to numerical instabilities. These instabilities limit the number of valid terms in the series, though the efficiency of time‐series integration even when resummation techniques like the Borel–Padé–Laplace (BPL) integrators are employed. In this study, we introduce a novel variational formulation for computing the terms of a TSE associated with a given PDE using higher‐order FEs. Our approach involves the incorporation of artificial diffusion terms on the left‐hand side of the equations corresponding to each power in the series, serving as a stabilization technique. We demonstrate that this method can be interpreted as a minimization of an energy functional, wherein the total variations of the unknowns are considered. Furthermore, we establish that the coefficients of the artificial diffusion for each term in the series obey a recurrence relation, which can be determined by minimizing the condition number of the associated linear system. We highlight the link between the proposed technique and the Discrete Maximum Principle (DMP) of the heat equation. We show, via numerical experiments, how the proposed technique allows having additional valid terms of the series that will be substantial in enlarging the stability domain of the BPL integrators.

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Performance of Borel–Padé–Laplace integrator for the solution of stiff and non-stiff problems

Cited by 7 publications

References 35 publications

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

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

Development of a hysteresis model based on axisymmetric and homotopic properties to predict moisture transfer in building materials

Stabilized time‐series expansions for high‐order finite element solutions of partial differential equations

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