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Ahmad Deeb

Publications and source records attributed to Ahmad Deeb.

9 recordsLinked to original sources
arXiv2026

Error estimation for numerical approximations of ODEs via composition techniques. Part II: BDF methods

Integration of Ordinary Differential Equations (ODEs) using Backward Difference formula (BDF) methods with p backward steps achieves order p accuracy if specific conditions are met. This work extends the composition technique with complex coefficients to the implicit BDF schemes, increasing the approximation order by one without additional backward points. The imaginary part of the composed flow provides an error estimate of order p + 1. Linear stability analysis reveals that the composed schemes break the Dahlquist barrier, achieving stability up to order eight. The computational performance of the composed flow outperforms BDF schemes when using the same number of backward points, allowing for higher accuracy with lower CPU time. For non-uniform meshes, the ratio of consecutive time steps, which influences stability, appears as a parameter in the roots of algebraic equations relative to the composed flow. Having a complex root with a real positive part implies a lower bound to this ratio depending on the order. For example, the bound is 0.4506 for order three and 0.6806 for order four. Numerical tests demonstrate the effectiveness of this technique in improving the accuracy and stability compared to BDF methods.

math.NA↗
arXiv2025

From Time Series Expansion to Proper Generalized Decomposition via Graph-Theoretical Connection: Stabilized Simulation of Fluids Flow

In this paper, we employ graph theory to establish a connection between the Time Series Expansion (TSE) and Proper Generalized Decomposition (PGD) methods. Using the concept of a directed graph, we demonstrate how one can transition from the computation of space modes in the TSE--first illustrated for the diffusion equation--to those of space modes in PGD, in which an inhomogeneous Volterra-type convolution recurrence relation, weighted by time-dependent coefficients, appears. This recurrence relation is simplified through graph-based analysis into a compact form using a simple path traversal, reducing the computational complexity. Moreover, the compact formulation reveals a natural stabilization process in the computation of space modes, where stabilized coefficients are automatically derived and can be used in the Stabilized-TSE (STSE) framework. To explicitly construct these coefficients, we consider a Simplified PGD (SPGD) formulation in which the time modes are chosen to be the time polynomial basis $t^n$. This choice yields a one-level Volterra-type recurrence relation that is similarly simplified using a simple path representation, demonstrating a connection in the computation of space modes from TSE, through STSE and SPGD, to PGD. This graph-based connection is exhibited in the case of inviscid flow to check how crucial the addition of an artificial diffusion is in stabilizing the recurrence formula of TSE. Finally, we extend the approach to the incompressible, dimensionless Navier-Stokes (NS) equations and build stabilization coefficients that depend on the Reynolds number Re, the space mode rank, and the simulation time step. Both the STSE and SPGD approaches are tested to simulate the wake behind a bluff body at Re = 5 000.

physics.flu-dyn↗
arXiv2025

Numerical Integration of Navier-Stokes Equations by Time Series Expansion and Stabilized FEM

This manuscript introduces an advanced numerical approach for the integration of incompressible Navier-Stokes (NS) equations using a Time Series Expansion (TSE) method within a Finite Element Method (FEM) framework. The technique is enhanced by a novel stabilization strategy, incorporating a Divergent Series Resummation (DSR) technique, which significantly augments the computational efficiency of the algorithm. The stabilization mechanism is meticulously designed to improve the stability and validity of computed series terms, enabling the application of the Factorial Series (FS) algorithm for series resummation. This approach is pivotal in addressing the challenges associated with the accurate and stable numerical solution of NS equations, which are critical in Computational Fluid Dynamics (CFD) applications. The manuscript elaborates on the variational formulation of Stokes problem and present convergence analysis of the method using the Ladyzhenskaya-Babuska-Brezzi (LBB) condition. It is followed by the NS equations and the implementation details of the stabilization technique, underscored by numerical tests on laminar flow past a cylinder, showcasing the method's efficacy and potential for broad applicability in fluid dynamics simulations. The results of the stabilization indicate a substantial enhancement in computational stability and accuracy, offering a promising avenue for future research in the field.

math.NA↗
arXiv2024

Error estimation for numerical approximations of ODEs via composition techniques. Part I: One-step methods

In this study, we introduce a refined method for ascertaining error estimations in numerical simulations of dynamical systems via an innovative application of composition techniques. Our approach involves a dual application of a basic one-step numerical method of order p in this part, and for the class of Backward Difference Formulas schemes in the second part [Deeb A., Dutykh D. and AL Zohbi M. Error estimation for numerical approximations of ODEs via composition techniques. Part II: BDF methods, Submitted, 2024]. This dual application uses complex coefficients, resulting outputs in the complex plane. The methods innovation lies in the demonstration that the real parts of these outputs correspond to approximations of the solutions with an enhanced order of p + 1, while the imaginary parts serve as error estimations of the same order, a novel proof presented herein using Taylor expansion and perturbation technique. The linear stability of the resulted scheme is enhanced compared to the basic one. The performance of the composition in computing the approximation is also compared. Results show that the proposed technique provide higher accuracy with less computational time. This dual composition technique has been rigorously applied to a variety of dynamical problems, showcasing its efficacy in adapting the time step,particularly in situations where numerical schemes do not have theoretical error estimation. Consequently, the technique holds potential for advancing adaptive time-stepping strategies in numerical simulations, an area where accurate local error estimation is crucial yet often challenging to obtain.

math.GM↗
arXiv2024

Stabilized Time Series Expansions for High-Order Finite Element Solutions of Partial Differential Equations

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-Pad\'e-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.

math.NA↗
arXiv2023

Numerical Approach Based on the Composition of One-Step Time-Integration Schemes For Highly Deformable Interfaces

In this work, we propose a numerical approach for simulations of large deformations of interfaces in a level set framework. To obtain a fast and viable numerical solution in both time and space, temporal discretization is based on the composition of one-step methods exhibiting higher orders and stability, especially in the case of stiff problems with strongly oscillatory solutions. Numerical results are provided in the case of ordinary and partial differential equations to show the main features and demonstrate the performance of the method. Convergence properties and efficiency in terms of computational cost are also investigated.

math.GM↗
arXiv2023

A Finite Element Approach For Modeling Biomembranes In Incompressible Power-Law Flow

We present a numerical method to model the dynamics of inextensible biomembranes in a quasi-Newtonian incompressible flow, which better describes hemorheology in the small vasculature. We consider a level set model for the fluid-membrane coupling, while the local inextensibility condition is relaxed by introducing a penalty term. The penalty method is straightforward to implement from any Navier-Stokes/level set solver and allows substantial computational savings over a mixed formulation. A standard Galerkin finite element framework is used with an arbitrarily high order polynomial approximation for better accuracy in computing the bending force. The PDE system is solved using a partitioned strongly coupled scheme based on Crank-Nicolson time integration. Numerical experiments are provided to validate and assess the main features of the method.

math.GM↗
arXiv2018

Some robust integrators for large time dynamics

This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through numerical examples. Next, Dirac integrators for constrained systems are exposed. An application on chaotic dynamics is presented. Lastly, for systems having no exploitable geometric structure, the Borel-Laplace integrator is presented. Numerical experiments on Hamiltonian and non-Hamiltonian systems are carried out, as well as on a partial differential equation. Keywords: Symplectic integrators, Dirac integrators, long-time stability, Borel summation, divergent series.

math.NA↗