Special Session 192: Numerical methods for complex differential equation models

A LAX-WENDROFF TYPE THEOREM OF DOUBLY CONSERVATIVE SCHEMES FOR DEGENERATE CONVECTION-DIFFUSION EQUATIONS

Juan Cheng Capital Normal University Peoples Rep of China Co-Author(s):

Abstract: It is well known that a numerical scheme for solving a hyperbolic conservation law, when convergent, may not always converge to a weak solution. The remedy is the famous Lax-Wendroff theorem, stating that a conservative numerical scheme, when convergent, always converges to a weak solution of the conservation law. In this talk, we address the same issue for numerically solving degenerate nonlinear diffusion or convection-diffusion equations. We start with an example showing that a conservative scheme, in the sense of that for conservation laws, may still converge to a function which is not a weak solution of the degenerate nonlinear diffusion equation. We then introduce a stronger form of conservative schemes, which we term as doubly-conservative (DoC) schemes, that would allow the proof of a Lax-Wendroff type theorem, namely if a DoC scheme converges, then it will converge to the weak solution of the degenerate nonlinear diffusion or convectiondiffusion equation. Finally, we design a new DoC local discontinuous Galerkin scheme that remains semi-discrete stable even when the diffusion coefficient degenerates. Numerical experiments demonstrate optimal convergence rates of this new scheme and validate our theoretical results.

Data-driven Discovery of Asymmetric Interacting Particle Systems

Jinchao Feng Great Bay University Peoples Rep of China Co-Author(s):    Sui Tang

Abstract: Interacting particle systems provide a powerful modeling framework for collective dynamics in nature and engineering. While prior methods have primarily addressed symmetric interactions using various learning techniques, many real-world systems exhibit asymmetric interactions, which demand more general and flexible modeling tools. In this talk, I will present a new Sparse Bayesian Learning (SBL) framework for identifying asymmetric interaction kernels in the Motsch-Tadmor model. By reformulating the nonlinear inverse problem as a subspace identification task, we establish identifiability guarantees and enable robust kernel recovery. Incorporating informative priors, the proposed SBL algorithm offers principled model selection and uncertainty quantification, achieving reliable inference from noisy trajectory data.

A Stabilized Numerical Framework for Necrotic Tumor Growth via Coupled Boundary Integral and Obstacle Solvers

Yu Feng Great Bay University Peoples Rep of China Co-Author(s):    Shuo Ling, Wenjun Ying, Zhennan Zhou

Abstract: We present a robust computational framework for Hele-Shaw tumor growth with necrotic cores, a problem identified as the incompressible limit of the Porous Media Equation. Simulating this system presents a fundamental challenge: while the outer boundary evolves via advection, the inner necrotic interface is defined by an obstacle problem and lacks an explicit advection structure, causing standard schemes to fail. To address this, we introduce a stabilized predictor-corrector strategy that iteratively resolves the bidirectional coupling between the nutrient-pressure fields and the domain geometry, ensuring robust time-stepping for both the advection-driven outer surface and the obstacle-defined necrotic core. We establish rigorous convergence theory for the single-interface case and demonstrate the method`s robustness in capturing the topological transition of necrotic core nucleation and complex geometric evolution.

Multi-fidelity numerical methods for a class of kinetic models

Liu Liu The Chinese University of Hong Kong Hong Kong Co-Author(s):

Abstract: In this talk, we will discuss about a class of multi-fidelity numerical methods for solving kinetic models. In the first part, we will address the uncertainty quantification for kinetic problems and development of bi-fidelity or tri-fidelity methods for different models, where some error estimates are studied using the hypocoercivity. In the second part, we will study an efficient asymptotic-preserving scheme for solving the Boltzmann equation with bi-fidelity algorithm designed in the velocity discretization. Lastly, some applications to deep learning approaches for kinetic models will be discussed, with the idea of bi-fidelity introduced. These are joint works with Xueyu Zhu, Lorenzo Pareschi, Nicolas Crouseilles, Zhen Hao.