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.

Explicit Symmetric Low-Regularity Integrator for the Nonlinear Schrodinger Equation

Yue Feng Xi`an Jiaotong University Peoples Rep of China Co-Author(s):

Abstract: The numerical approximation of low-regularity solutions to the nonlinear Schrodinger equation (NLSE) is notoriously difficult and even more so if structure-preserving schemes are sought. Recent works have been successful in establishing symmetric low-regularity integrators for NLSE. However, so far, all prior symmetric low-regularity algorithms are fully implicit, and therefore require the solution of a nonlinear equation at each time step, leading to significant numerical cost in the iteration. In this work, we introduce the first fully explicit (multi-step) symmetric low-regularity integrators for NLSE. We demonstrate the construction of an entire class of such schemes which notably can be used to symmetrise (in explicit form) a large amount of existing low-regularity integrators. We provide rigorous convergence analysis of our schemes and numerical examples demonstrating both the favourable structure preservation properties obtained with our novel schemes, and the significant reduction in computational cost over implicit methods.

The unified gas kinetic wave-particle method for the neutron transport equation

Yanli Wang Beijing Computational Science Research Center Peoples Rep of China Co-Author(s):    Guangwei Liu, Shuang Tan

Abstract: The unified gas-kinetic wave-particle (UGKWP) method is proposed for the neutron transport equation, addressing the inherent multiscale nature of neutron propagation in both optically thin and thick regimes. UGKWP couples macroscopic diffusion and microscopic transport processes within a unified time-dependent framework, allowing a smooth transition between the free transport and diffusion regimes. This method is readily extended to multi-group neutron transport models and is applicable to both steady-state and eigenvalue problems. Several numerical examples, including the 1D and 3D single-group and 3D multi-group problems, are studied, indicating UGKWP a promising framework for scalable and accurate simulation of multigroup neutron transport in complex geometries.