arXiv subjects

Ruiyang Dai

Publications and source records attributed to Ruiyang Dai.

6 recordsLinked to original sources

arXiv2026

On the Gram matrix of standard inner products of asymmetrically-weighted Hermite functions

Let A denote the infinite Gram matrix associated with the standard L2 inner product of asymmetrically-weighted (AW) Hermite functions. We derive an explicit representation of its entries and its Cholesky factorization. We further show that this factorization admits a natural interpretation on a scaled Bargmann-Fock basis. An explicit formula for the inverse of A is also obtained. We then consider the corresponding finite Gram matrix and analyze its asymptotic property, as well as that of its Schur complement. The analysis is motivated by numerical methods for plasma physics, in particular Galerkin spectral methods applied to the Vlasov-Poisson (VP) system. As an application, we demonstrate how the derived Gram matrix formulas and asymptotic results can be exploited in the analysis and implementation of a Galerkin spectral method for the VP system.

math.NA↗

arXiv2026

A stable and efficient Galerkin spectral method with asymmetrically-weighted Hermite functions for the Vlasov-Poisson system

We analyze a Galerkin spectral method applied to the Vlasov-Poisson (VP) system based on time-independent asymmetrically-weighted (AW) Hermite functions in velocity. The VP system is written as an hyperbolic system using AW Hermite functions in velocity. Unlike the classical Petrov-Galerkin spectral method, which is not stable, the proposed Galerkin spectral method admits a natural stability in the unweighted L2 norm. This stability property enables a rigorous convergence analysis of the method. For sufficiently regular solutions with exponential decay in velocity, we establish error estimates between the exact and numerical solutions and prove convergence of the Galerkin spectral method. In particular, the method achieves spectral convergence in Sobolev spaces, with convergence rates determined by the regularity of the exact solution. Since the Gram matrix arising from the proposed Galerkin method is dense, we derive an equivalent form that retains the same approximation properties while preserving the sparsity structure similar to the classical Petrov-Galerkin method.

math.NA↗

arXiv2026

Galerkin method for asymmetrically-weighted Hermite approximations applied to the Vlasov-Poisson system

We investigate a numerical method for the Vlasov-Poisson (VP) system utilizing asymmetrically-weighted (AW) Hermite bases in velocity space, which is an hyperbolic system. In particular, we concentrate on spectral methods in velocity. For the Hermite spectral form of the VP system, we analyze the resaon that the form with AW Hermite bases can be instable. To obtain L2 stability properties, we consider a Galerkin method intead of the classical Petrov-Galerkin method, which naturally ensures stability with respect to the L2 norm. We also present an equivalent form of the method that maintains a computational cost modest compared to that of the Petrov-Galerkin method. Finally, we present numerical simulations based on the proposed Hermite spectral method, showcasing its stability.

math.NA↗

arXiv2024

On the quadratic stability of asymmetric Hermite basis with application to plasma physics with oscillating electric field

We analyze why the discretization of linear transport with asymmetric Hermite basis functions can be instable in quadratic norm. The main reason is that the finite truncation of the infinite moment linear system looses the skew-symmetry property with respect to the Gram matrix. Then we propose an original closed formula for the scalar product of any pair of asymmetric basis functions. It makes possible the construction of two simple modifications of the linear systems which recover the skew-symmetry property. By construction the new methods are quadratically stable with respect to the natural $L^2$ norm. We explain how to generalize to other transport equations encountered in numerical plasma physics. Basic numerical tests with oscillating electric fields of different nature illustrate the unconditional stability properties of our algorithms.

math.NA↗

arXiv2023

A Block Jacobi Sweeping Preconditioner for the Helmholtz Equation

In recent research, the parallel performances of sweeping-type algorithms for high-frequency time-harmonic wave problems have been improved by departing from standard layer-type domain decomposition and introducing a new sweeping strategy on a checkerboard-type domain decomposition, where sweeps can be performed more flexibly. These sweeps can be done by a certain number of steps, each of which provides the necessary information from subdomains solved at the current iteration to their next neighboring subdomains. Although, subproblems in these subdomains can be solved concurrently at each step, the sequential nature of the process of the sweeping approaches still exists, which limits their potential for parallelization. We propose a block Jacobi sweeping preconditioner, which is an improved variant of sweeping-type preconditioners. The new feature of these improved variants can be interpreted as several partial sweeps, which can be thought of as sweeps that operate on a subset of the subdomains in parallel. We present several two-dimensional finite element results to study and compare the sweeping preconditioner and the block Jacobi sweeping preconditioner.

math.NA↗

arXiv2021

Multidirectionnal sweeping preconditioners with non-overlapping checkerboard domain decomposition for Helmholtz problems

This paper explores a family of generalized sweeping preconditionners for Helmholtz problems with non-overlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on high-order transmission conditions and cross-point treatments, which cannot scale without an efficient preconditioning technique when the number of subdomains increases. With the proposed approach, existing sweeping preconditioners, such as the symmetric Gauss-Seidel and parallel double sweep preconditioners, can be applied to checkerboard partitions with different sweeping directions (e.g. horizontal and diagonal). Several directions can be combined thanks to the flexible version of GMRES, allowing for the rapid transfer of information in the different zones of the computational domain, then accelerating the convergence of the final iterative solution procedure. Several two-dimensional finite element results are proposed to study and to compare the sweeping preconditioners, and to illustrate the performance on cases of increasing complexity.

math.NA↗