arXiv subjects

Hamad El Kahza

Publications and source records attributed to Hamad El Kahza.

3 recordsLinked to original sources

arXiv2026

A Structure-Preserving Penalization Method for the Single-species Rosenbluth-Fokker-Planck Equation

The Rosenbluth-Fokker-Planck (RFP) equation describes Coulomb collisional dynamics within and across species in plasmas. It belongs to the broader class of anisotropic-diffusion-advection equations, whose numerical approximation is highly-nontrivial due to its nonlinearity, stiffness, and structural properties such as conservation and entropy dissipation (hence with the Maxwellian distribution as the equilibrium state). In this paper, we propose a structure-preserving penalization scheme for the stiff, single-species RFP equation. The scheme features three novel components: 1) a novel generalization of the well-known Chang-Cooper discretization for the RFP equation that is equilibrium-preserving and enables positivity while preserving mass, momentum, and energy; 2) an easy-to-invert isotropic variable-coefficient penalization operator to deal with the temporal stiffness without resorting to a fully implicit scheme, borrowing ideas from explicit-implicit-null (EIN) methods, and 3) an adaptive timestepping strategy that preserves the positivity of the full penalized scheme. The resulting scheme conserves mass, momentum, and energy strictly, is unconditionally stable, and robustly positivity preserving. The scheme is demonstrated with linear and nonlinear anisotropic diffusion examples of increasing complexity, including several single-species RFP examples.

math.NA↗

arXiv2025

Sylvester-Preconditioned Adaptive-Rank Implicit Time Integrators for Advection-Diffusion Equations with Variable Coefficients

We consider the adaptive-rank integration of {2D and 3D} time-dependent advection-diffusion partial differential equations (PDEs) with variable coefficients. We employ a standard finite-difference method for spatial discretization coupled with diagonally implicit Runge-Kutta temporal schemes. The discrete equation is a generalized Sylvester equation (GSE), which we solve with an adaptive-rank algorithm structured around three key strategies: {(i) constructing dimension-wise subspaces based on an extended Krylov strategy, (ii) developing an effective preconditioner for the reduced coefficient matrix, and (iii) efficiently computing the residual of the equation without explicitly reverting to the full-rank form. {The low-rank decomposition is performed in 2D using SVD, and with high-order SVD (HOSVD) in 3D to represent the tensor in a compressed Tucker format.} The computational complexity of the proposed approach {is demonstrated numerically to} be comparable to the constant-coefficient case [El Kahza et al, J. Comput. Phys., 518 (2024)], scaling as $\mathcal{O}(N {r^2} + {r^{d+1}})$ for $d$-dimensional problems (here, $d = 2$ or $3$), with $N$ the resolution in one dimension and $r$ the maximal rank during the Krylov iteration (which we find to be largely independent of $N$). We present numerical examples that illustrate the computational efficacy and complexity of our algorithm.}

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arXiv2024

Krylov-based Adaptive-Rank Implicit Time Integrators for Stiff Problems with Application to Nonlinear Fokker-Planck Kinetic Models

We propose a high order adaptive-rank implicit integrators for stiff time-dependent PDEs, leveraging extended Krylov subspaces to efficiently and adaptively populate low-rank solution bases. This allows for the accurate representation of solutions with significantly reduced computational costs. We further introduce an efficient mechanism for residual evaluation and an adaptive rank-seeking strategy that optimizes low-rank settings based on a comparison between the residual size and the local truncation errors of the time-stepping discretization. We demonstrate our approach with the challenging Lenard-Bernstein Fokker-Planck (LBFP) nonlinear equation, which describes collisional processes in a fully ionized plasma. The preservation of {the equilibrium state} is achieved through the Chang-Cooper discretization, and strict conservation of mass, momentum and energy via a Locally Macroscopic Conservative (LoMaC) procedure. The development of implicit adaptive-rank integrators, demonstrated here up to third-order temporal accuracy via diagonally implicit Runge-Kutta schemes, showcases superior performance in terms of accuracy, computational efficiency, equilibrium preservation, and conservation of macroscopic moments. This study offers a starting point for developing scalable, efficient, and accurate methods for high-dimensional time-dependent problems.

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