Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.01593

Energy-Dissipative, High-Order BDF Schemes for Simulating Geometric Flows via Energy Increments

Zhiqing Pan·Wei Jiang·Jiwei Jia
View original source ↗Read source document ↗Export metadata ↓

Abstract

We propose energy-dissipative backward differentiation formula (BDF) methods for geometric flows in the Barrett--Garcke--Nürnberg (BGN) parametric finite element framework. Starting from a Lagrange-multiplier formulation, we replace the multistep approximation of the energy derivative by an energy-increment equation while retaining the BDF geometric equations. The Simpson quadrature and two interleaved time grids provide third- and fourth-order constructions. The positive quadrature weights ensure stepwise decay of the original discrete geometric energy for every nondegenerate solution of the nonlinear system. We establish an explicit conditional consistency bound for the scalar energy equation and a regularity criterion for the Newton system of curve-shortening flows. Then, the proposed schemes extend to simulating curve diffusion, mean curvature flow, and surface diffusion. Finally, numerical experiments demonstrate geometric accuracy, energy decay, good mesh distribution, and high performance of the proposed schemes.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhiqing Pan, Wei Jiang, Jiwei Jia. 2026-11-01. Energy-Dissipative, High-Order BDF Schemes for Simulating Geometric Flows via Energy Increments. https://arxiv.org/abs/2611.01593

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

KEEP EXPLORING

Related papers

arXiv2026

Extracting Dual Analytic Geometries of Linear Transformations to Achieve Efficient Computation

We generalize the Calderon-Zygmund methodology for operator analysis and the Rokhlin-Greengard fast multipole algorithms, in which the domain and range of an operator are jointly organized. Our methods analyze unstructured kernels by uncovering the joint hidden geometries of their domain and range. The main tool is the Questionnaire algorithm, an iterative procedure that constructs adaptive bi-hierarchical partition trees that reveal latent multiscale organization and local low-rank structure in the kernel. When the geometry is unknown, the iteration starts from any initial affinity, even a random one, and, with a stochastic flip perturbation and a spectral space-filling curve that fixes the left-right order of every tree, reorganizes the matrix into an image of the operator's "phase space": the joint space of its sources and sensors, in duality, as positions and frequencies are for the Fourier transform. This ordering gives the phase space an analytic structure on which any classical transform can be carried out; the resulting compression is direct evidence of that structure. Guided by these geometries, we obtain efficient factorizations and fast matrix-vector products in two ways: (1) the Butterfly algorithm, which exploits the learned bi-hierarchical localized low-rank structure; and (2) the adaptive eGHWT, which selects best tilings in space and sequency from all levels of generalized Haar-Walsh wavelet packets. We call the resulting algorithms the Questionnaire Factorization and Fast Transform (QFFT). Unlike classical approaches requiring prior knowledge of the geometry, QFFT is fully data-driven and applies to matrices from irregular or unknown distributions. We demonstrate it on heterogeneous oscillatory operators and on orthogonal eigenvector expansions with high multiplicity and random sampling, reducing storage from O(N^2) to O(N log N) and enabling fast, scalable computation.

math.NA↗
arXiv2026

Decoupled energy-stable Runge-Kutta schemes of arbitrary order for the anisotropic phase-field dendritic crystal growth model

This paper introduces a novel and general framework for constructing arbitrary-order, energy-stable, and decoupled schemes for the anisotropic phase-field dendritic crystal growth model. The proposed framework combines a time-dependent auxiliary variable, an algebraically stable Runge-Kutta method, and a matrix diagonalization technique. This unique combination yields a scheme that is proven to be unconditionally energy-dissipative while requiring only the solution of independent, constant-coefficient elliptic equations at each time step, which can be solved in parallel. The auxiliary variable is recovered from a uniquely solvable $q\times q$ algebraic system. We provide a rigorous proof of $q$th-order convergence in time for a $q$-stage Runge-Kutta method within a fixed Fourier-Galerkin space. Numerical experiments in 2D and 3D not only confirm the theoretical convergence and energy stability but also demonstrate the efficiency of the decoupled approach and provide a detailed investigation of key parameters' effects on dendritic morphology.

math.NA↗
arXiv2026

Automated Numerical Stability Analysis of Deep Learning Operators

Finite-precision arithmetic unavoidably introduces numerical approximation errors. Numerical computations may use insufficient precision or an improper formulation, which leads to numerical instability. In this paper, we introduce a unified software tool for stochastic numerical validation of deep-learning operators. The tool follows CESTAC on supported exposed operations and uses an operator-level data-perturbation approximation for GEMM-like kernels. Our developed software not only enables numerical validation with a single computation pass but also detects the sources of numerical instability and provides numerical stability monitoring during deep learning training and inference. We verified its effectiveness on the detection of polluted operators with injected numerical instabilities across various tasks. We believe that our developed method and tools provide valuable insights into developing numerically stable computing kernels, which are particularly critical for numerically stable and efficient deep learning training and inference.

math.NA↗