Articles
Publication Date: 2026
Journal of Computational and Applied Mathematics (03770427)482
This paper presents a vector-valued moving least squares (MLS) approximation for reconstructing vector fields that are divergence-free or curl-free. The proposed method constructs analytically divergence-free or curl-free shape functions by applying appropriate differential operators to a stream (potential) function approximated using the MLS method. The procedure involves solving a sparse linear least squares problem, for which the Conjugate Gradient Least Squares (CGLS) algorithm is employed to reduce computational costs compared to the direct solvers. The approach relies solely on a set of scattered nodes in the computational domain and requires no background triangulation. We provide error bounds for the approximation and support the theoretical bounds with numerical experiments. Additionally, we demonstrate the application of the divergence-free MLS approximation to the numerical solution of Darcy's flow equations. © 2026 The Author(s)
Publication Date: 2026
Computer Physics Communications (0010-4655)319
The classical Landau-Lifshitz-Gilbert (LLG) equation has long served as a cornerstone for modeling magnetization dynamics in magnetic systems, yet its classical nature limits its applicability to inherently quantum phenomena such as entanglement and nonlocal correlations. Inspired by the need to incorporate quantum effects into spin dynamics, recently a quantum generalization of the LLG equation is proposed [Phys. Rev. Lett. 133, 266704 (2024)] which captures essential quantum behavior in many-body systems. In this work, we develop a robust numerical methodology tailored to this quantum LLG framework that not only handles the complexity of quantum many-body systems but also preserves the intrinsic mathematical structures and physical properties dictated by the equation. We apply the proposed method to a class of quantum systems with a moderate number of spins that host host topological states of matter, and demonstrate rich quantum behavior, including the emergence of long-time entangled states. This approach opens a pathway toward reliable simulations of quantum magnetism beyond classical approximations, potentially leading to new discoveries. © 2025 The Author(s)
Publication Date: 2024
Journal of Computational Physics (219991)508
In this research article, we introduce a high-order and non-oscillatory finite volume method in combination with radial basis function approximations and use it for the solution of scalar conservation laws on unstructured meshes. This novel approach departs from conventional non-oscillatory techniques, which often require the use of multiple stencils to achieve smooth reconstructions. Instead, the new method uses a single central stencil and hinges on an approximate interpolation methodology called the weighted smoothed reconstruction (WSR), with a foundation on polyharmonic spline interpolation. Through some numerical experiments, we demonstrate the efficiency and accuracy of the new approach. It reduces the computational cost and performs well in capturing shocks and sharp solution fronts. © 2024 The Author(s)
Publication Date: 2023
Numerical Algorithms (10171398)93(4)pp. 1759-1794
In this paper, we present a scattered data approximation method for detecting and approximating the discontinuities of a bivariate function and its gradient. The new algorithm is based on partition of unity, polyharmonic kernel interpolation, and principal component analysis. Localized polyharmonic interpolation in partition of unity setting is applied for detecting a set of fault points on or close to discontinuity curves. Then a combination of partition of unity and principal component regression is used to thinning the detected points by moving them approximately on the fault curves. Finally, an ordered subset of these narrowed points is extracted and a parametric spline interpolation is applied to reconstruct the fault curves. A selection of numerical examples with different behaviors and an application for solving scalar conservation law equations illustrate the performance of the algorithm. © 2023, The Author(s).
Publication Date: 2023
JOURNAL OF COMPUTATIONAL PHYSICS (00219991)479
In this paper a new direct RBF partition of unity (D-RBF-PU) method is developed for numerical solution of partial differential equations defined on smooth orientable surfaces which are discretized with sets of scattered nodes and with approximations to normal vectors at each of the nodes. The accuracy, stability and efficiency of the new method are studied through some theoretical and experimental results. This method is a localized RBF based technique, results in a perfectly sparse final linear system, uses only scattered nodes on the surface rather than a connected mesh, and is applicable for a large class of PDEs on manifolds. Applications to some biological and chemical reaction-diffusion models are also given. Results show that the new method outperforms other comparable techniques for surface PDEs.(c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).
