Special Session 39: Recent Developments in Gradient Flows: Theory, Numerics, and Applications

Dynamics in some phase field models

Yuan Chen Chinese University of Hong Kong(Shenzhen) Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will introduce some dynamics arising from some phase field models. Theoretical mathematical analysis including sharp interface limit and existence of solutions will be presented.

Adaptive time-stepping and maximum-principle-preserving flow dynamic approach for gradient flows

Qing Cheng Tongji University Peoples Rep of China Co-Author(s):    Qianqian Liu, Wenbin Chen, Jie Shen

Abstract: We develop an adaptive time-stepping approach for gradient flows with distinct treatments for conservative and non-conservative dynamics. For the non-conservative gradient flows in Lagrangian coordinates, we propose a modified formulation augmented by auxiliary terms to guarantee positivity of the determinant, and prove that the corresponding adaptive second-order Backward Difference Formulas (BDF2) scheme preserves energy stability and the maximum principle under suitable time-step ratio constraint.

Sharp interface limits of some phase field models

Mingwen Fei Anhui Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk we shall discuss sharp interface limits of some diffuse interface models. This is done by constructing an approximate solution from the limiting flow via matched asymptotic expansions, and then estimating its difference with the real solution which is based on a spectrum inequality of the linearized operator at the transition profile (also called optimal profile).

Super-convergent HDG methods for the Cahn-Hilliard equation

Daozhi Han State University of New York at Buffalo USA Co-Author(s):    Gang Chen, Jiaxuan Liu, John Singler, Yangwen Zhang, Dujin Zuo

Abstract: We present and analyze a super-convergent hybridizable discontinuous Galerkin method for solving the Cahn-Hilliard equation. The HDG method utilizes polynomials of degree k+1 for scalar variables and polynomials of degree k for fluxes and numerical traces with reduced diffusion stabilization. For the classical Cahn-Hilliard equation we establish optimal convergence rates for all variables and all polynomial orders with error constants depending on the inverse of interface thickness in polynomial orders. For the advective Cahn-Hilliard equation we show that optimal convergence can be obtained without any advection stabilization. The key tools involved include the HDG spectral estimate of the linearized Cahn-Hilliard operator and new elliptic projection adapted to advection.

A gradient flow model for ground state calculations in wigner formalism based on density functional theory

Guanghui Hu University of Macau Macau Co-Author(s):    Li Ruo; Zhan Hongfei

Abstract: In this talk, a gradient flow model is presented for conducting ground state calculations in the Wigner formalism of many-body systems in the framework of density functional theory. Theoretically, an energy functional in the Wigner formalism is proposed, based on which the minimization problem is designed and analyzed for the ground state, providing a new perspective for ground state calculations of the Wigner function. Employing density functional theory, a gradient flow model is built upon the energy functional to obtain the ground state Wigner function representing the entire many-body system. Numerically, a parallelizable algorithm is developed using the operator splitting method and Fourier spectral collocation method. Numerical experiments demonstrated the anticipated accuracy, encompassing the one-dimensional system with upto 2^21 particles and the three dimensional system with defect, showcasing the potential of our approach towards the large scale simulations.

A generalized SAV approach for nonlinear dissipative systems

Fukeng Huang Eastern Institute of Technology, Ningbo Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will present the generalized SAV approach, which possessed the following attractive properties:1)It yields a long time uniform upper bound for the numerical solutions unconditionally; 2) It is applicable to general dissipative systems, not limited to gradient flow systems; 3)It enables the construction of arbitrarily high-order schemes with low computational cost. As an application, we construct the longtime stable schemes for the forced Navier-Stokes equations. Numerical examples will be presented to demonstrate the advantages of our approach.

The Cahn-Hilliard equation with a source term

Alain Miranville University Le Havre Normandie France Co-Author(s):

Abstract: Our aim in this talk is to discuss the Cahn-Hilliard equation with a (nonlinear) source term and a logarithmic nonlinear term. Such an equation has applications in, e.g., tumor growth and image processing. In particular, we discuss the existence of weak solutions and their regularity.

Mixed finite element methods for biharmonic obstacle problems

Paolo Piersanti The Chinese University of Hong Kong, Shenzhen Peoples Rep of China Co-Author(s):    Tianyu Sun

Abstract: This talk introduces a mixed Finite Element Method aimed at approximating solutions to fourth-order variational problems with constraints. We first address the biharmonic obstacle problem and propose an error convergence framework that offers an alternative to the established approaches by Ciarlet & Raviart and Ciarlet & Glowinski. Our method emphasises improved ease of numerical implementation, which may enhance practical usability. Next, we investigate a two-dimensional variational problem involving linearly elastic shallow shells constrained within a specified half-space. We begin with cases where the middle surface has non-zero curvature and demonstrate that applying a mixed Finite Element Method with conforming elements requires a symmetry constraint on the gradient matrix of the dual variable. Notably, we find that this implementation cannot rely solely on Courant triangles, indicating a variation in approach based on the geometric features of the problem. This constitutes a counterexample to the statement that solutions of fourth order linear problems can be approximated by solely resorting to Courant triangles if one considers the mixed formulation of the original problem. We notice that this counterexample arises in connection with the lack of rigidity of linearly elastic shallow shells middle surface. In cases where the middle surface is flat, we observe that the symmetry constraint is not necessary, allowing for the use of Courant triangles alone for solution approximation. This observation suggests that shell geometry can significantly influence the selection of Finite Element methods for discretization. Our theoretical findings are supported by a series of supplementary numerical experiments, illustrating the practicality of the proposed methods.

Structure-preserving compact splitting methods for Schr\\odinger-type equations

Jie Shen Eastern Institute of Technology, Ningbo Peoples Rep of China Co-Author(s):

Abstract: We develop a novel class of high-order, structure-preserving methods for Schr\{o}dinger-type equations based on a predictor--corrector framework that integrates operator splitting with Lagrange multiplier corrections. The predictor employs high-order compact splitting spectral methods, in which the Hamiltonian is decomposed into two parts that are exactly solvable in either phase or physical space, while the corrector enforces the exact preservation of multiple invariants through Lagrange multipliers. We rigorously prove the existence and uniqueness of the Lagrange multipliers under reasonable conditions, and show that the overall scheme retains the same temporal order as the underlying predictor. The resulting schemes constitute the first class of split-type schemes capable of preserving multiple original invariants, and they demonstrate markedly improved long-time accuracy and stability compared to classical splitting approaches. Numerical experiments validate the theoretical findings and illustrate the effectiveness of the proposed methods for rotating dipolar Bose--Einstein condensates with two invariants and rotating spin-1 systems with three invariants. More broadly, the proposed framework provides a simple and general strategy for systematically transforming existing time integrators into structure-preserving methods for Hamiltonian systems

High-order energy stable numerical schemes for gradient flows

Xiaoming Wang Eastern Institute of Technology Peoples Rep of China Co-Author(s):    Wenbin Chen, Zhaohui Fu, Haifeng Wang, Shufeng Wang

Abstract: We present several high-order energy stable numerical schemes for gradient flows by combining exponential time differencing with either multi-step (MS) or Runge-Kutta (RK) method. All schemes are variable-step and unconditionally stable. Numerical experiments demonstrate the effectiveness of these schemes in both fixed-step and adaptive-step setting.

Adaptive feature capture method for solving partial differential equations with near singular solutions

Xiaoping Wang The Chinese University of Hong Kong (Shenzhen) /Shenzhen International Center for Industrial and Applied Mathematics Peoples Rep of China Co-Author(s):    Xiaoping Wang

Abstract: We propose the Adaptive Feature Capture Method (AFCM), a novel machine learning framework that adaptively redistributes neurons and collocation points in high gradient regions to enhance local expressive power. AFCM employs the gradient norm of an approximate solution as a monitor function to guide the reinitialization of feature function parameters. This ensures that partition hyperplanes and collocation points cluster where they are most needed, achieving higher resolution without increasing computational overhead.