arXiv subjects

Qinghai Zhang

Publications and source records attributed to Qinghai Zhang.

17 recordsLinked to original sources

arXiv2026

A multiphase cubic MARS method for fourth- and higher-order interface tracking of two or more materials with arbitrary topology and geometry

For interface tracking of an arbitrary number of materials in two dimensions, we propose a multiphase cubic MARS method that (a) represents the topology and geometry of the interface via graphs, cycles, and cubic splines, (b) applies to any number of materials with arbitrarily complex topology and geometry, (c) maintains an $(r,h)$-regularity of the interface so that the distance between any pair of adjacent markers is within a user-specified range, (d) distributes the markers adaptively along the interface so that arcs with high curvature are resolved by densely populated markers, and (e) achieves fourth-, sixth-, and eighth-order accuracy both in time and in space.} In particular, all possible types of junctions, which pose challenges to VOF methods and level-set methods, are handled with ease. Results of a variety of benchmark tests confirm the analysis and demonstrate the superior accuracy, efficiency, and versatility of the proposed method.

math.NA↗

arXiv2026

A Fourth-Order Cut-cell Multigrid Method for Solving Elliptic Equations on Arbitrary Domains

To numerically solve a generic elliptic equation on two-dimensional domains with rectangular Cartesian grids, we propose a cut-cell geometric multigrid method that features (1) general algorithmic steps that apply to two-dimensional constant-coefficient elliptic equations with both divergence and non-divergence forms and all types of boundary conditions, (2) the versatility of handling both regular and irregular domains with arbitrarily complex topology and geometry, (3) the fourth-order accuracy even at the presence of ${\cal C}^1$ discontinuities on the domain boundary, and (4) the optimal complexity of $O(h^{-2})$.Test results demonstrate the generality, accuracy, efficiency, robustness, and excellent conditioning of the proposed method.

math.NA↗

arXiv2025

A linear MARS method for three-dimensional interface tracking

For explicit interface tracking in three dimensions, we propose a linear MARS method that (a) represents the interface by a partially ordered set of glued surfaces and approximates each glued surface with a triangular mesh, (b) maintains an $(r,h,θ)$-regularity on each triangular mesh so that the distance between any pair of adjacent markers is within the range $[rh,h]$ and no angle in any triangle is less than $θ$, (c) applies to three-dimensional continua with arbitrarily complex topology and geometry, (d) preserves topological structures and geometric features of moving phases under diffeomorphic and isometric flow maps, and (e) achieves second-order and third-order accuracy in terms of the Lagrangian and Eulerian length scales, respectively. Results of classic benchmark tests verify the effectiveness of the novel mesh adjustment algorithms in enforcing the $(r,h,θ)$-regularity and demonstrate the high accuracy and efficiency of the proposed linear MARS method.

math.NA↗

arXiv2025

On Topology of Three-dimensional Continua with Singular Points

We propose to model the topology of three-dimensional (3D) continua by Yin sets, regular open semianalytic sets with bounded boundary. Our model differs from manifold-based models in that singular points of a 3D continuum, i.e., boundary points where the tangent plane is not uniquely defined, are treated not as anomalies but as a central subject of our theoretical investigation. We characterize the local and global topology of Yin sets. Then we give a unique boundary representation of Yin sets based on the notion of a glued surface, a quotient space of an orientable compact 2-manifold along a one-dimensional CW complex. Our results apply to 3D continua with arbitrarily complex topology and may be useful in a number of scientific and engineering applications such as solid modeling, computer-aided design, and numerical simulations of multiphase flows with topological changes.

math.GT↗

arXiv2025

On Convergence of the Secant Method

The secant method, as an important approach for solving nonlinear equations, is introduced in nearly all numerical analysis textbooks. However, most textbooks only briefly address the Q-order of convergence of this method, with few providing rigorous mathematical proofs. This paper establishes a rigorous proof for the Q-order of convergence of the secant method and theoretically compares its computational efficiency with that of Newton's method.

math.NA↗

arXiv2025

GePUP-ES: High-order Energy-stable Projection Methods for the Incompressible Navier-Stokes Equations with No-slip Conditions

Inspired by the unconstrained PPE (UPPE) formulation [Liu, Liu, & Pego 2007 Comm. Pure Appl. Math., 60 pp. 1443], we previously proposed the GePUP formulation [Zhang 2016 J. Sci. Comput., 67 pp. 1134] for numerically solving the incompressible Navier-Stokes equations (INSE) on no-slip domains. In this paper, we propose GePUP-E and GePUP-ES, variants of GePUP that feature (a) electric boundary conditions with no explicit enforcement of the no-penetration condition, (b) equivalence to the no-slip INSE, (c) exponential decay of the divergence of an initially non-solenoidal velocity, and (d) monotonic decrease of the kinetic energy. Different from UPPE, the GePUP-E and GePUP-ES formulations are of strong forms and are designed for finite volume/difference methods under the framework of method of lines. Furthermore, we develop semi-discrete algorithms that preserve (c) and (d) and fully discrete algorithms that are fourth-order accurate for velocity both in time and in space. These algorithms employ algebraically stable time integrators in a black-box manner and only consist of solving a sequence of linear equations in each time step. Results of numerical tests confirm our analysis.

math.NA↗

arXiv2025

Fourth-order Adaptive Mesh Refinement both in space and in time for incompressible Navier-Stokes equations with Dirichlet boundary conditions

We present a fourth-order projection method with adaptive mesh refinement (AMR) for numerically solving the incompressible Navier-Stokes equations (INSE) with subcycling in time. Our method features (i) a reformulation of INSE so that the velocity divergence decays exponentially on the coarsest level, (ii) a derivation of coarse-fine interface conditions that preserves the decay of velocity divergence on any refinement level of the AMR hierarchy, (iii) an approximation of the coarse-fine interface conditions via spatiotemporal interpolations to facilitate subcycling in time, (iv) enforcing to machine precision solvability conditions of elliptic equations over each connected component of the subdomain covered by any refinement level, (v) a composite projection for synchronizing multiple levels, and (vi) geometric multigrid for solving linear systems with optimal complexity. Different from current block-structured AMR algorithms, our method never adopts refluxing at the coarse-fine interface, nor is fine-to-coarse averaging applied to projected velocities. Results of numerical tests demonstrate the high accuracy and efficiency of the proposed method.

math.NA↗

arXiv2025

Fourth- and Higher-Order Semi-Lagrangian Finite Volume Methods for the Two-dimensional Advection Equation on Arbitrarily Complex Domains

To numerically solve the two-dimensional advection equation, we propose a family of fourth- and higher-order semi-Lagrangian finite volume (SLFV) methods that feature (1) fourth-, sixth-, and eighth-order convergence rates, (2) applicability to both regular and irregular domains with arbitrarily complex topology and geometry, (3) ease of handling both zero and nonzero source terms, and (4) the same algorithmic steps for both periodic and incoming penetration conditions. Test results confirm the analysis and demonstrate the accuracy, flexibility, robustness, and excellent conditioning of the proposed SLFV method.

math.NA↗

arXiv2025

Fourth- and higher-order finite element methods for the incompressible Navier-Stokes equations with Dirichlet boundary conditions

Inspired by the unconstrained pressure Poisson equation (PPE) formulation [Liu, Liu, \& Pego, Comm. Pure Appl. Math. 60 (2007): 1443-1487], we previously proposed the generic projection and unconstrained PPE (GePUP) formulation [Zhang, J. Sci. Comput. 67 (2016): 1134-1180] for numerically solving the incompressible Navier-Stokes equations (INSE) with no-slip boundary conditions. In GePUP, the main evolutionary variable does not have to be solenoidal with its divergence controlled by a heat equation. This work presents high-order finite-element solvers for the INSE under the framework of method-of-lines. Continuous Lagrange finite elements of equal order are utilized for the velocity and pressure finite element spaces to discretize the weak form of GePUP in space, while high-order implicit-explicit Runge-Kutta methods are then employed to treat the stiff diffusion term implicitly and the other terms explicitly. Due to the implicit treatment of the diffusion term, the time step size is only restricted by convection. The solver is efficient in that advancing the solution at each time step only involves solving a sequence of linear systems either on the velocity or on the pressure with geometric multigrid methods. Furthermore, the solver is enhanced with adaptive mesh refinement so that the multiple length scales and time scales in flows at moderate or high Reynolds numbers can be efficiently resolved. Numerical tests with various Reynolds numbers are performed for the single-vortex test, the lid-driven cavity, and the flow past a cylinder/sphere, demonstrating the high-order accuracy of GePUP-FEM both in time and in space and its capability of accurately and efficiently capturing the right physics. Moreover, our solver offers the flexibility in choosing velocity and pressure finite element spaces and is free of the standard inf-sup condition.

math.NA↗

arXiv2025

Lagrangian Particle Classification and Lagrangian Flux Identities for a Moving Hypersurface

For a moving hypersurface in the flow of a nonautonomous ordinary differential equation in $n$-dimensional Euclidean spaces, the fluxing index of a passively-advected Lagrangian particle is the total number of times it crosses the moving hypersurface within a time interval. The problem of Lagrangian particle classification is to decompose the phase space into flux sets, equivalence classes of Lagrangian particles at the initial time. In the context of scalar conservation laws, the problem of Lagrangian flux calculation (LFC) is to find flux identities that relate the Eulerian flux of a scalar through the moving hypersurface, a spatiotemporal integral over the moving surface in a given time interval, to spatial integrals over donating regions at the initial time of the interval. In this work, we implicitly characterize flux sets via topological degrees, explicitly construct donating regions, prove the equivalence of flux sets and donating regions, and establish two flux identities; these analytical results constitute our solutions to the aforementioned problems. Based on a flux identity suitable for numerical calculation, we further proposed a new LFC algorithm, proved its convergence, and demonstrated its efficiency, good conditioning, and high-order accuracy by results of various numerical tests.

math.NA↗

arXiv2025

A Fast Fourth-Order Cut Cell Method for Solving Elliptic Equations in Two-Dimensional Irregular Domains

We propose a fast fourth-order cut cell method for solving constant-coefficient elliptic equations in two-dimensional irregular domains. In our methodology, the key to dealing with irregular domains is the poised lattice generation (PLG) algorithm that generates finite-volume interpolation stencils near the irregular boundary. We are able to derive high-order discretization of the elliptic operators by least squares fitting over the generated stencils. We then design a new geometric multigrid scheme to efficiently solve the resulting linear system. Finally, we demonstrate the accuracy and efficiency of our method through various numerical tests in irregular domains.

math.NA↗

arXiv2025

A fourth-order cut-cell method for solving the two-dimensional advection-diffusion equation with moving boundaries

We propose a fourth-order cut-cell method for solving the two-dimensional advection-diffusion equation with moving boundaries on a Cartesian grid. We employ the ARMS technique to give an explicit and accurate representation of moving boundaries, and introduce a cell-merging technique to overcome discontinuities caused by topological changes in cut cells and the small cell problem. We use a polynomial interpolation technique base on poised lattice generation to achieve fourth-order spatial discretization, and use a fourth-order implicit-explicit Runge-Kutta scheme for time integration. Numerical tests are performed on various moving regions, with advection velocity both matching and differing from boundary velocity, which demonstrate the fourth-order accuracy of the proposed method.

math.NA↗

arXiv2024

MetaRuleGPT: Recursive Numerical Reasoning of Language Models Trained with Simple Rules

Recent studies have highlighted the limitations of large language models in mathematical reasoning, particularly their inability to capture the underlying logic. Inspired by meta-learning, we propose that models should acquire not only task-specific knowledge but also transferable problem-solving skills. We introduce MetaRuleGPT, a novel Transformer-based architecture that performs precise numerical calculations and complex logical operations by learning and combining different rules. In contrast with traditional training sets, which are heavily composed of massive raw instance data, MetaRuleGPT is pre-trained on much less abstract datasets containing basic, compound, and iterative rules for mathematical reasoning. Extensive experimental results demonstrate MetaRuleGPT can mimic human's rule-following capabilities, break down complexity, and iteratively derive accurate results for complex mathematical problems. These findings prove the potential of rule learning to enhance the numerical reasoning abilities of language models.

cs.CL↗

arXiv2024

A fourth-order, multigrid cut-cell method for solving Poisson's equation in three-dimensional irregular domains

We propose a fourth-order cut-cell method for solving Poisson's equations in three-dimensional irregular domains. Major distinguishing features of our method include (a) applicable to arbitrarily complex geometries, (b) high order discretization, (c) optimal complexity. Feature (a) is achieved by Yin space, which is a mathematical model for three-dimensional continua. Feature (b) is accomplished by poised lattice generation (PLG) algorithm, which finds stencils near the irregular boundary for polynomial fitting. Besides, for feature (c), we design a modified multigrid solver whose complexity is theoretically optimal by applying nested dissection (ND) ordering method.

math.NA↗

arXiv2024

An AI-aided algorithm for multivariate polynomial reconstruction on Cartesian grids and the PLG finite difference method

Polynomial reconstruction on Cartesian grids is fundamental in many scientific and engineering applications, yet it is still an open problem how to construct for a finite subset $K$ of $\mathbb{Z}^{\textsf{D}}$ a lattice $\mathcal{T}\subset K$ so that multivariate polynomial interpolation on this lattice is unisolvent. In this work, we solve this open problem of poised lattice generation (PLG) via an interdisciplinary research of approximation theory, abstract algebra, and artificial intelligence (AI). Specifically, we focus on the triangular lattices in approximation theory, study group actions of permutations upon triangular lattices, prove an isomorphism between the group of permutations and that of triangular lattices, and dynamically organize the AI state space of permutations so that a depth-first search of poised lattices has optimal efficiency. Based on this algorithm, we further develop the PLG finite difference method that retains the simplicity of Cartesian grids yet overcomes the disadvantage of legacy finite difference methods in handling irregular geometries. Results of various numerical tests demonstrate the effectiveness of our algorithm and the simplicity, efficiency, and fourth-order accuracy of the PLG finite difference method.

math.NA↗

arXiv2022

A fourth-order unfitted characteristic finite element method for solving the advection-diffusion equation on time-varying domains

We propose a fourth-order unfitted characteristic finite element method to solve the advection-diffusion equation on time-varying domains. Based on a characteristic-Galerkin formulation, our method combines the cubic MARS method for interface tracking, the fourth-order backward differentiation formula for temporal integration, and an unfitted finite element method for spatial discretization. Our convergence analysis includes errors of discretely representing the moving boundary, tracing boundary markers, and the spatial discretization and the temporal integration of the governing equation. Numerical experiments are performed on a rotating domain and a severely deformed domain to verify our theoretical results and to demonstrate the optimal convergence of the proposed method.

math.NA↗

arXiv2019

Fluid Modeling and Boolean Algebra for Arbitrarily Complex Topology in Two Dimensions

We propose a mathematical model for fluids in multiphase flows in order to establish a solid theoretical foundation for the study of their complex topology, large geometric deformations, and topological changes such as merging. Our modeling space consists of regular open semianalytic sets with bounded boundaries, and is further equipped with constructive and algebraic definitions of Boolean operations. Major distinguishing features of our model include (a) topological information of fluids such as Betti numbers can be easily extracted in constant time, (b) topological changes of fluids are captured by non-manifold points on fluid boundaries, (c) Boolean operations on fluids correctly handle all degenerate cases and apply to arbitrarily complex topologies, yet they are simple and efficient in that they only involve determining the relative position of a point to a Jordan curve and intersecting a number of curve segments. Although the main targeting field is multiphase flows, our theory and algorithms may also be useful for related fields such as solid modeling, computational geometry, computer graphics, and geographic information system.

math.AT↗