Schedule for: 25w5344 - Structure Preserving Schemes for Complex Nonlinear Systems

A set dinner is served daily between 5:30pm and 7:30pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

We study non-conservative hyperbolic systems of balance laws and are interested in development of well-balanced (WB) numerical methods for such systems. One of the ways to enforce the balance between the flux terms and source and non-conservative product terms is to rewrite the studied system in a quasi-conservative form by incorporating the latter terms into the modified global flux. The resulting system can be quite easily solved by Riemann-problem-solver-free central-upwind (CU) schemes. This approach, however, does not allow to accurately treat non-conservative products. We therefore apply a path-conservative (PC) integration technique and develop a very robust and accurate path-conservative central-upwind schemes (PCCU) based on flux globalization. I will demonstrate the performance of the WB PCCU schemes on a wide variety of examples.

In this talk, I will present our recent results on high-order methods for general dissipative systems, including the Navier–Stokes and Cahn–Hilliard equations. The proposed high-order schemes are built upon two key techniques: (1) the generalized scalar auxiliary variable (SAV) approach, and (2) the generalized backward differentiation formula (BDF) methods. We will report both the numerical schemes and the associated error analyses.

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

Classical numerical schemes often rely on polynomial bases to approximate solutions, primarily due to their analytical simplicity and well-understood properties. Accordingly, most structure-preserving discretizations have been developed within polynomial approximation frameworks. However, polynomial spaces may be ill-suited for accurately representing solutions in certain regimes,  for example in boundary layers, where exponential functions offer superior approximation properties. In this talk, we focus on the construction of  structure-preserving numerical schemes based on general (non-polynomial) function spaces. Therefore, we introduce the concept of Function-spaces summation-by-parts operators and present the current-state-of-art using them to solve advection-dominated problems.

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.

Kinetic models describe a wide range of physical processes relevant to natural and engineering sciences where a large number of particles is involved. Compared to macroscopic models such as classical fluid dynamics equations, kinetic models lie closer to particle descriptions. More detail to specific phenomena and a more accurate mathematical description of the problem may be achieved by kinetic models in particular in the areas of radiative transfer, rarefied gases and turbulence modelling. For small scaling parameters, kinetic models generally converge to a macroscopic limit model. Therefore, asymptotic preserving schemes are desired which allow to pass from the kinetic to the macroscopic model on the discrete level. However, the construction of schemes satisfying this property is not trivial. In addition, kinetic equations often result in stiff problems. In this talk, we discuss upwind SBP schemes, a class of schemes which relax SBP properties by allowing for the inclusion of upwind dissipation, and their combination with implicit-explicit time discretization in order to construct efficient and stable asymptotic preserving numerical schemes for kinetic equations.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

As technology advances, our exploration of nature has reached increasingly extreme conditions and scales, heightening the demand for accurate and robust numerical solutions of complex nonlinear differential equations. A fundamental challenge lies in preserving the intrinsic physical structures of these equations—such as positivity, boundedness, divergence-free properties, and entropy dissipation—during numerical computation. Maintaining these structures is essential for ensuring the stability and reliability of simulations. In this talk, I will present some of our recent effort on structure-preserving numerical methods for solving partial differential equations and modeling unknown equations. Topics will include structure-preserving schemes for hyperbolic PDEs and their applications in fluid dynamics and astrophysics, as well as data- and physics-informed approaches for learning unknown governing equations.

This paper introduces a high-order discontinuous Galerkin (DG) scheme that simultaneously ensures positivity-preserving (PP) properties and maintains the globally divergence-free (GDF) constraint for the ideal magnetohydrodynamics (MHD) equations. Achieving both conditions remains quite challenging in MHD simulations when the PP and GDF reconstructions (theoretically linked) are treated as independent post-processing operations. To overcome this difficulty, we propose the pointwise total energy correction method, which is easy to implement and highly effective. Specifically, based on the equation of state, we recompute the total energy values at integration nodes using PP hydrodynamic variables and the GDF magnetic field. As long as the GDF magnetic field is highly accurate, the total energy obtained through this correction technique can also maintain high accuracy. The key technique for proving the PP property of the GDF-DG scheme involves the convex decomposition of cell averages. We carefully selected numerical integration formulas based on Gauss-Lobatto quadrature points. Moreover, the corrected total energy values at these points can uniquely interpolate a piecewise $\mathcal{Q}^{k+1}$ polynomial. With the geometric quasilinearization framework introduced by Wu et al., our GDF-DG scheme can be theoretically shown to satisfy the PP property. To suppress spurious oscillations, we adopt the jump filter for ideal MHD equations, as a sequel to our recent work. This filter operates locally based on jump information at cell interfaces, and preserves key attributes of the DG method, such as conservation, $L^2$ stability, and high-order accuracy. Moreover, it boasts an impressively low computational cost, given that no local characteristic decomposition is required and all computations are confined to the physical space. The jump filter is applied after each Runge-Kutta stage without altering the DG spatial discretization and maintains both the GDF and PP properties of the scheme. Numerical simulations demonstrate the accuracy, effectiveness, and robustness of the proposed PP-GDF-DG schemes with the jump filter. 

We introduce novel first- and second-order numerical schemes for simulating convection in coupled free-flow and porous media systems governed by the Navier–Stokes–Darcy–Boussinesq equations. The methods integrate a mean-reverting Scalar Auxiliary Variable (SAV) formulation with BDF time discretization and are designed for high computational efficiency—requiring only the solution of three linear, symmetric positive-definite, constant-coefficient problems per time step. Importantly, the schemes preserve the system’s long-time dissipation structure, ensuring stability over extended integrations. Preliminary numerical simulations highlight their effectiveness in capturing transitions between deep and shallow convection, a key feature in geophysical and engineering contexts. We also outline the broader applicability of the approach to geophysical fluid models aimed at capturing long-time statistical behavior (i.e., climate).

The focus of this talk is on space-time discretizations of kinetic transport equations, which are used for example in the study of neutron transport. In particular, we consider a linear transport equation under a diffusive scaling with periodic boundary conditions, which converges to the heat equation with a source term when the Knudsen number tends to zero. The goal is to formulate stable space-time discretizations of the problem that are also asymptotic preserving, so that the numerical scheme converges to an approximation of the underlying macroscopic PDE. For this purpose, we employ summation-by-parts (SBP) operators to approximate the spatial and temporal derivatives in the equations, and simultaneous approximation terms (SATs) to impose boundary and initial conditions. We prove that the schemes are stable and asymptotic preserving, and conclude by considering some numerical simulations.

In the talk we will discuss a diffuse-interface (phase-field) model for tumor growth that takes into account the nutrient consumption and chemotaxis. For this tumor growth model described by a nonlinear system consisting of a Cahn--Hilliard-type equation coupled with a reaction-diffusion equation, we constructed an efficient scheme based on the idea of the scalar auxiliary variable, which we show are not only decoupled and easy to implement, but also have the properties of mass conservation and unconditional energy stability. Furthermore, we derive rigorous error estimates for the tumor and nutrient variables of the scheme. Several numerical examples are presented to validate the accuracy and stability of the proposed scheme. This is a joint work with Zhaoyang Wang.

Surface water waves of practical relevance, such as tsunamis and solitary waves, are inherently nonlinear and dispersive. Although the governing equations derived from first principles, namely Euler’s equations, provide a complete description of wave propagation, they remain extremely challenging to analyze and simulate directly. As a result, numerous approximate systems have been introduced in water wave theory. Among these, we identify a system of partial differential equations that has been rigorously shown to be well-posed in the sense of Hadamard, even in bounded domains with slip-wall boundary conditions and variable bottom topography. This system also satisfies three fundamental conservation laws - mass, energy, and a less common but essential irrotationality conservation law - and remains consistent with the Euler equations for large time intervals. To solve this system numerically, we developed a modified Galerkin/Finite Element Method for the space discretization, accompanied by a relaxation Runge-Kutta method, designed to preserve all three conservation laws at the discrete level. This structure-preserving approach not only ensures numerical stability but also enhances the physical fidelity of the simulations. The properties of the new numerical method are analysed while its accuracy and robustness are validated through numerical simulations and comparison with laboratory experimental data.

Existing neural network models to learn Hamiltonian systems, such as SympNets, although accurate in low-dimensions, struggle to learn the correct dynamics for high-dimensional many-body systems. Herein, we introduce Symplectic Graph Neural Networks (SympGNNs) that can effectively handle system identification in high-dimensional Hamiltonian systems, as well as node classification. SympGNNs combines symplectic maps with permutation equivariance, a property of graph neural networks. Specifically, we propose two variants of SympGNNs: i) G-SympGNN and ii) LA-SympGNN, arising from different parameterizations of the kinetic and potential energy. We demonstrate the capabilities of SympGNN on two physical examples: a 40-particle coupled Harmonic oscillator, and a 2000-particle molecular dynamics simulation in a two-dimensional Lennard-Jones potential. Furthermore, we demonstrate the performance of SympGNN in the node classification task, achieving accuracy comparable to the state-of-the-art. We also empirically show that SympGNN can overcome the oversmoothing and heterophily problems, two key challenges in the field of graph neural networks.

Entropy stable discontinuous Galerkin (DG) methods improve the robustness of high order DG simulations of nonlinear conservation laws. These methods yield a semi-discrete entropy inequality, and rely on an algebraic flux differencing formulation which involves both summation-by-parts (SBP) discretization matrices and entropy conservative two-point finite volume fluxes. However, explicit expressions for such two-point finite volume fluxes may not be available for all systems, or may be computationally expensive to compute. We propose an alternative approach to constructing entropy stable DG methods using an artificial viscosity coefficient based on the local violation of a cell entropy inequality and a local entropy dissipation estimate. The resulting method yields the same global semi-discrete entropy inequality satisfied by entropy stable flux differencing DG methods. The artificial viscosity coefficients are parameter-free and locally computable over each cell. The resulting artificial viscosity preserves high order accuracy, improves linear stability, and does not result in a more restrictive maximum stable time-step size under explicit time-stepping.

In the current stage of numerical methods for PDE, the primary challenge lies in addressing the complexities of high dimensionality while maintaining physical fidelity in our solvers. In this presentation, I will introduce deep learning assisted particle methods aimed at addressing some of these challenges. These methods combine the benefits of traditional structure-preserving techniques with the approximation power of neural networks, aiming to handle high dimensional problems with minimal training. I will begin with a discussion of general Wasserstein-type gradient flows and then extend the concept to the Landau equation in plasma physics.

We introduce novel entropy-dissipative numerical schemes for a class of kinetic equations, leveraging the recently introduced scalar auxiliary variable (SAV) approach. Both first and second order schemes are constructed. Since the positivity of the solution is closely related to entropy, we also propose positivity-preserving versions of these schemes to ensure robustness, which include a scheme specially designed for the Boltzmann equation and a more general scheme using Lagrange multipliers. The accuracy and provable entropy-dissipation properties of the proposed schemes are validated for both the Boltzmann equation and the Landau equation through extensive numerical examples. Based on the joint work with Shiheng Zhang and Jie Shen.

Robust and efficient numerical simulation of high-Mach-number compressible turbulence remains a formidable challenge due to intricate shock-turbulence interactions. While significant progress has been made for system governed by the Euler equations, validated results based on the more physically accurate Navier-Stokes framework remain scarce. Our numerical investigations reveal that the dominant computational difficulty arises from the extreme stiffness of viscous diffusion terms in near-vacuum regions, where conventional explicit time-marching schemes frequently fail. To address this, we propose a novel numerical framework combining two key components: (1) a low-diffusion positivity-preserving discretization for convective fluxes to maintain physical realizability, and (2) a semi-implicit approach coupled with domain decomposition techniques to stabilize the stiff viscous terms. Benchmark tests demonstrate that our methodology effectively suppresses unphysical oscillations caused by vacuum-induced viscosity stiffness, a critical limitation of conventional explicit formulations. Application to the isotropic compressible turbulence simulation with unprecedented turbulent Mach number (Ma_t > 8) confirms the scheme's capability to achieve high-fidelity solutions while maintaining numerical stability across multi-scale flow regimes.

In this talk, we introduce generalized solutions of compressible flows, the so-called dissipative solutions. We will concentrate on inviscid flows—the Euler equations—and also mention relevant results obtained for viscous compressible flows governed by the Navier–Stokes equations. Dissipative solutions are obtained as limits of suitable structure-preserving, consistent, and stable finite volume schemes [1–4]. When a strong solution to the governing equations exists, the dissipative weak solution coincides with it on its lifespan [1]. Otherwise, we apply K-convergence and prove strong convergence of the empirical means of numerical solutions to a dissipative weak solution [5,6]. The latter is the expected value of the dissipative measure-valued solutions and satisfies a weak formulation of the Euler equations modulo the Reynolds turbulent stress. If time permits, we will also derive error estimates for the corresponding finite volume method. The error analysis is carried out via the relative energy, which serves as a problem-suited "metric" [4]. Theoretical results will be illustrated by a series of numerical simulations. References [1] E. Feireisl, M. Lukáčová-Medvid'ová, H. Mizerová, B. She (2021). Numerical analysis of compressible fluid flows, Springer. [2] E. Feireisl, M. Lukáčová-Medvid'ová, H. Mizerová (2020). Convergence of finite volume schemes for the Euler equations via dissipative-measure-valued solutions. Found. Comput. Math., 20, 923–966. [3] E. Feireisl, M. Lukáčová-Medvid'ová, H. Mizerová, B. She (2019). Convergence of a finite volume scheme for the compressible Navier–Stokes system. ESAIM: Math. Model. Numer. Anal., 53, 1957–1979. [4] M. Lukáčová-Medvid'ová, B. She, Y. Yuan (2022). Error estimate of the Godunov method for multidimensional compressible Euler equations. J. Sci. Comput., 91, 71. [5] E. Feireisl, M. Lukáčová-Medvid'ová, H. Mizerová (2020). K-convergence as a new tool in numerical analysis. IMA J. Numer. Anal., 40, 2227–2255. [6] E. Feireisl, M. Lukáčová-Medvid'ová, B. She, Y. Wang (2021). Computing oscillatory solutions of the Euler system via K-convergence. Math. Models Methods Appl. Sci., 31, 537–576.

In this talk, we present a strongly mass-conservative numerical scheme for the coupled Navier–Stokes and Darcy–Forchheimer problem, governed by mass conservation, balance of normal forces, and the Beaver–Joseph–Saffman (BJS) interface condition. The proposed method employs a staggered discontinuous Galerkin (DG) approach for the Navier–Stokes equations, guaranteeing a divergence-free velocity field, while a mixed finite element formulation is used to solve the Darcy–Forchheimer problem. Crucially, the scheme enforces interface conditions directly without using Lagrange multipliers, which ensures the strong mass conservation across the whole computational domain. Under suitable small data assumptions, we establish the well-posedness of the formulation and derive a rigorous convergence analysis. By carefully addressing the interplay between interface terms and discretization, we prove optimal convergence rates for all variables, with velocity and velocity gradient errors independent of pressure approximation, demonstrating the scheme's pressure-robustness. Numerical experiments will be presented to confirm the theoretical results.

We propose energy-conserving numerical schemes for the Vlasov-type  systems. Those equations are fundamental models in the simulation of plasma  physics. The total energy is an important physical quantity that is conserved  by those models. Our methods are the first Eulerian solver that can  preserve fully discrete total energy conservation. The main features of our  methods include energy-conservative temporal and spatial discretization. We validate our  schemes by rigorous derivations and benchmark numerical examples.

This talk introduces the recently developed Cell-Average-Based Neural Network (CANN). This method is inspired by finite volume schemes and replaces traditional spatial and temporal discretization with a learned explicit one-step method. The well-trained parameters of the network act as the coefficients for the scheme. Unlike conventional numerical methods, the CANN approach is not limited by small time step CFL conditions, enabling the use of significantly larger time steps. This leads to a highly efficient and rapid computational method. We present a conservative version of the CANN method designed for nonlinear conservation laws. This conservative approach ensures mass conservation and effectively captures relevant physical solutions, including contact discontinuities, shock collisions, and interactions between shocks and rarefaction waves. Additionally, we will discuss recent results related to the bound-preserving neural network method, which maintains L-infinity stability for the piecewise constant numerical solution.

Dispersive PDEs, such as Klein-Gordon equation, Dirac equation, Schrodinger equation, arise from many different areas, e.g. computational chemistry, plasma physics, quantum mechanics. Typical computational tasks in dispersive PDEs are finding the ground/stationary states and solving the dynamics. In this talk, we report some recent advances on the numerical methods and analysis for the time-dependent dispersive PDEs, paying particular attention to the highly oscillatory PDEs, which usually exhibit solutions with high frequency waves in time and/or in space, and are generally computational expensive.

In this talk, we present a highly efficient and accurate time-splitting scheme for Vlasov-type equations in the quasi-neutral regime. The proposed scheme exhibits several remarkable properties: (i) The nonlinear subproblem of the time-splitting scheme is analytically solvable, and the numerical solutions of each subproblem are free of time-discretization error; (ii) It preserves the oscillation frequency of the plasma system, the wave-form in time of the electric field and the distribution function, as well as the asymptotic limit in the quasi-neutral regime; (iii) It strictly conserves the mass and energy; It is worth- while to point out that the analytical expressions of the nonlinear subproblem do not rely on the asymptotic limit of the Vlasov equations, and therefore this scheme can be used to solve general plasma systems without being restricted to the limit regime. Extensive numerical experiments including the 3D3V case have been conducted to show the accuracy, efficiency, and robustness of the proposed scheme.

Abstract: In this talk, we construct high-order energy dissipative and conservative local discontinuous Galerkin methods for the generalized nonlinear dispersive equations, which can be regarded as the Hamilton system and have infinitely many conservative quantities. We give the proofs for the dissipation and conservation for several related conservative quantities. The corresponding semi-discrete error estimates can be obtained. The capability of our schemes for different types of solutions is shown via several numerical experiments. We extend our work to the space-time Galerkin method to explore the theoretical analysis in fully discrete schemes.

Strong form collocation methods over scattered node layouts allow for easy construction of high-order methods, but do not naturally provide energy stability for time-dependent PDEs. We show that the lack of stability can be viewed as arising from inexact quadrature together with discontinuities of the trial space functions in applicable cases. With this knowledge, we can control these issues to achieve a stable scheme with small energy losses.