arXiv subjects

Tingfeng Wang

Publications and source records attributed to Tingfeng Wang.

6 recordsLinked to original sources

arXiv2026

Some uniform error analysis for gKdV equations in dispersionless limit regime before dispersive shock

This work establishes uniform error estimates for classical numerical schemes applied to the generalized Korteweg-de Vries (gKdV) equation in the dispersionless limit regime, specifically before the development of dispersive shock wave (DSW). We analyze the Crank-Nicolson (CN) and a Lawson-type Runge-Kutta (Lawson-RK) methods, when discretized in space via the Fourier pseudo-spectral method. We prove that both fully discrete schemes achieve optimal second-order temporal accuracy and spectral spatial accuracy, with error constants being uniform in the vanishing dispersion parameter $\varepsilon$. The analysis also addresses the unique solvability of the CN scheme in the dispersionless limit regime. Theoretical findings are supported by numerical experiments, demonstrating the $\varepsilon$-uniform accuracy and the necessity of step size restriction for CN. The study in general validates the classical methods for solving the dispersionless gKdV equation before DSW.

math.NA↗

arXiv2026

Analysis of gradient flow for computing defocusing action ground states of rotating nonlinear Schr\"odinger equations

This work focuses on the numerical computation of defocusing action ground states for rotating nonlinear Schr\"odinger equations (RNLS) using a direct gradient flow (DGF) method. We address theoretical gaps in the existing literature concerning the stability and convergence of this DGF scheme. Firstly, we prove the unconditional stability of the DGF scheme, demonstrating that the action functional is monotonically non-increasing along the discrete flow for arbitrary time step sizes. Secondly, we establish a rigorous convergence analysis, proving global convergence under minor assumptions and local exponential convergence to the action ground state under a reasonable non-degeneracy condition. The analysis relies on the uniform boundedness of sublevel sets of the action functional and introduces a tailored $H^1$-distance between phase-shift equivalence classes to handle complex-valued ground states with quantized vortices. A novel analytical framework is also developed to establish the exponential convergence rate. Numerical experiments are presented to validate the theoretical findings, demonstrating both the global migration towards a neighborhood of the ground state and subsequent exponential convergence.

math.NA↗

arXiv2026

Convergence analysis of $L^{p+1}$-normalized gradient flow for action ground state of nonlinear Schr\"odinger equation

This paper presents a rigorous convergence analysis of the $L^{p+1}$-normalized gradient flow with asymptotic Lagrange multiplier (GFALM) method for computing the action ground state of the nonlinear Schr\"odinger equation in the focusing case. First, a general global convergence theory is established for the semi-discrete GFALM scheme, guaranteeing the existence of an accumulation point and a convergent subsequence. Then, under additional non-degeneracy assumptions, a local exponential convergence rate is rigorously proven. This result is further extended to the fully discrete case using a Fourier pseudo-spectral discretization. The analysis is achieved by characterizing the local geometry of the $L^{p+1}$-constrained manifold near the ground state, establishing a quadratic growth property of the energy functional, and deriving a \L{}ojasiewicz-type gradient inequality. Finally, the paper also investigates the exponential convergence of the associated continuous-time gradient flow, providing a theoretical foundation for future numerical discretizations. This work extends existing convergence analyses for energy ground states, addressing the challenges posed by the $L^{p+1}$ constraint, especially the absence of an inner-product structure.

math.NA↗

arXiv2025

SafeGRPO: Self-Rewarded Multimodal Safety Alignment via Rule-Governed Policy Optimization

Multimodal large language models (MLLMs) have demonstrated impressive reasoning and instruction-following capabilities, yet their expanded modality space introduces new compositional safety risks that emerge from complex text-image interactions. Such cross-modal couplings can produce unsafe semantics even when individual inputs are benign, exposing the fragile safety awareness of current MLLMs. While recent works enhance safety by guiding models to reason about potential risks, unregulated reasoning traces may compromise alignment; although Group Relative Policy Optimization (GRPO) offers self-rewarded refinement without human supervision, it lacks verifiable signals for reasoning safety. To address this, we propose SafeGRPO a self-rewarded multimodal safety alignment framework that integrates rule-governed reward construction into GRPO, enabling interpretable and verifiable optimization of reasoning safety. Built upon the constructed SafeTag-VL-3K dataset with explicit visual, textual, and combined safety tags, SafeGRPO performs step-guided safety thinking to enforce structured reasoning and behavior alignment, substantially improving multimodal safety awareness, compositional robustness, and reasoning stability across diverse benchmarks without sacrificing general capabilities.

cs.CR↗

arXiv2025

Arbitrary High-Order Maximum Principle-Preserving and Energy Dissipating Schemes for Gradient Flows

For gradient flows, the existing structure-preserving schemes are difficult to achieve arbitrary high-order accuracy in time while preserving maximum-principle (MBP) and energy dissipating simultaneously. In this paper, we develop a new framework for constructing structure-preserving schemes which shall preserve those nice properties. By introducing KKT-conditions for energy dissipating and bound-preserving, we rewrite the original gradient flow into an expanded and coupled system. We shall utilize a novel predictor-corrector-corrector framework, termed the PCC method, which consists of a prediction from any numerical scheme to the user's favor, followed by two correction steps designed to enforce energy stability and MBP, respectively. We take the exponential time differencing Runge-Kutta scheme (ETDRK) as an example and establish the unique solvability and robust error analysis for our new framework. Extensive numerical experiments are provided to validate the efficiency and accuracy of our new approach. Enough numerical comparisons with the existing popular schemes are shown that our structure-preserving schemes can avoid numerical oscillations and capture the exact evolution of energy.

math.NA↗

arXiv2024

Improved scalar auxiliary variable schemes for original energy stability of gradient flows

Scalar auxiliary variable (SAV) methods are a class of linear schemes for solving gradient flows that are known for the stability of a `modified' energy. In this paper, we propose an improved SAV (iSAV) scheme that not only retains the complete linearity but also ensures rigorously the stability of the original energy. The convergence and optimal error bound are rigorously established for the iSAV scheme and discussions are made for its high-order extension. Extensive numerical experiments are done to validate the convergence, robustness and energy stability of iSAV, and some comparisons are made.

math.NA↗