Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1504.02881

Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime

Weizhu Bao·Yongyong Cai·Xiaowei Jiao·Qinglin Tang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size $h$ and time step $τ$ as well as the small parameter $\varepsilon$. Based on the error bounds, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the FDTD methods share the same $\varepsilon$-scalability on time step: $τ=O(\varepsilon^3)$. Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability on time step is improved to $τ=O(\varepsilon^2)$ when $0<\varepsilon\ll 1$. Extensive numerical results are reported to support our error estimates.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Weizhu Bao, Yongyong Cai, Xiaowei Jiao, Qinglin Tang. 2016-02-21. Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime. https://doi.org/10.1007/s10915-016-0333-3

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Stability of Block Eliminations and Additive Modifications

The block elimination with additive modifications (BEAM) method was recently proposed as a alternative to LU with partial pivoting requiring less communication. Because of the novelty of BEAM, the existing theoretical analysis is lacking. To that end, we analyze both the numerical stability of the underlying block LU factorization and the effects of additive modifications. For the block LU factorization, we are able to improve the previous results of Demmel et al. from being cubic in the element growth to merely quadratic. Furthermore, we propose an alternative measure of element growth that is better aligned with block LU; this new measure of growth allows our analysis to apply to matrices that cannot be factored with pointwise LU. In the second part, we analyzed the modifications produced by BEAM and the effect they have on the condition number and growth factor. Finally, we show that BEAM will not apply any modifications in some cases that regular block LU can safely factor.

math.NA↗
arXiv2026

Efficient Rigorous Continuation via Chebyshev Series Expansion I

We study the global continuation of solution manifolds arising in dynamical systems. We present a rigorous continuation method based on a Chebyshev series expansion of the solution manifold. The branch is first approximated by a high-order Chebyshev interpolation polynomial, and an explicit error bound is then obtained by verifying the contraction of a quasi-Newton operator near this approximation. The contraction is formulated on a weighted $\ell^1$ space, giving a finer control than the typical $C^0$-error bound obtained from the uniform contraction theorem. In fact, the latter follows directly from our contraction operator. Furthermore, we discuss how our strategy applies naturally to pseudo-arclength continuation, where the continuation parameter fails to provide a valid local coordinate, and extends to multi-parameter continuation. Lastly, we detail two applications in which we compute a two-parameter family of steady-states for the Cahn--Hilliard equation, and a one-parameter family of steady-states undergoing saddle-node bifurcations for the Shigesada--Kawasaki--Teramoto system.

math.NA↗
arXiv2026

Efficient iterative techniques for solving tensor problems with the T-product

This paper develops two efficient iterative methods for solving tensor equations under the T-product framework. For T-symmetric positive definite tensor equations of the form $\mathcal{C} \star \mathcal{X} = \mathcal{D}$, we propose a conjugate-gradient-type algorithm that generates orthogonal residual and $\mathcal{C}$-orthogonal direction sequences, ensuring convergence within a finite number of steps. For general consistent tensor equations, we extend the method using a normal-equation transformation, and further adapt it to handle inconsistent systems by solving a least-squares minimization problem. Key advantages include direct tensor-based computations without explicit matrix expansion, rigorous finite-step convergence proofs, and the ability to obtain minimal Frobenius norm solutions. Numerical experiments on synthetic data, benchmark images, and video sequences demonstrate that the proposed algorithms achieve high precision with low computational time, confirming their practicality for large-scale multidimensional problems.

math.NA↗