2025/12/06-2025/12/08
Remi Abgrall, University of Zurich, Switzerland
Tilte：The PAMPA Algorithm: Bound Preservation, Boundary Conditions and Non Oscillatory Properties
Abstract: In this talk, we will present further improvement of the PAMPA Algorithm which is a variant of the active flux method. More precisely, in the active flux method, due to Roe and co-authors, the time space discretisation are linked via the method of characteristics or some exact evolution operator, hence the name of the method because the flux a part of the solution. Here, we use the method of lines and we focus on the space approximation of the solution, in one and two dimensions, using unstructured or polygonal meshes.
After a short introduction of the method which is a combination of finite volume and finite difference approximation that are interlaced in a particular manner, we will first present a simple bound preserving algorithm. This method is not specific to the Pampa algorithm. However, the method has two difficulties: first being bound preserving does not mean that we are oscillation free, and second, handling the boundary condition is a bit obscure because part of the algorithm is inspired by finite difference. We will show how to solve both issues, and will provide several numerical examples that show the behaviour of the new algorithm.
The authors acknowledge several inspiring and important discussions with Professor Kailiang Wu, SUSTech.
Wasilij Barsukow, University of Bordeaux, France
Title: Structure-Preservation Properties of Active Flux Methods on Cartesian Grids
Abstract: The Active Flux method evolves cell averages and point values, which are situated on cell interfaces and shared among adjacent cells. The evolution of the averages is virtually unambiguous, but that of the point values can be achieved in many different ways. Starting from the formulation employing evolution operators I will show that Active Flux is stationarity preserving for linear acoustics when the exact evolution operator is used. I will then show that a semi-discrete point value update naturally arises from a Petrov-Galerkin framework when using biorthogonal test and basis functions. Although this semi-discrete method does not use an exact update, it still turns out to be stationarity preserving for linear acoustics. I will show that this property is, in fact, linked to the nature of the approximation space rather than the particular way of updating the point values. This explains the good behaviour of Active Flux at low Mach number for the Euler equations.
Alina Chertock, NC State Univerity, USA
Title: Developing Asymptotic Preserving Schemes via Nonconservative Formulation
Abstract: In this talk, we present a new numerical method for solving the compressible Euler equations that is reliable for both low- and high-Mach-number flows. When the Mach number is small, the equations become stiff, and standard explicit schemes require very small time steps, making them inefficient. Our goal is to design an asymptotic-preserving (AP) scheme that remains accurate and stable across all Mach number regimes.
Our method takes a different route from traditional AP approaches, which typically rely on conservative flux splitting. Instead, we rewrite the equations in primitive (nonconservative) variables and apply a nonconservative hyperbolic splitting. We then discretize the stiff part using second-order implicit central differences, while the nonstiff part is handled with an explicit second-order central-upwind scheme. A key component of the method is solving a Poisson-type elliptic equation for the pressure at each time level, which ensures the AP property. At the same time, we evolve the conservative form of the equations using a semi-discrete central-upwind scheme. After each time step, a post-processing step selects which numerical solution to use based on the Mach number: the AP nonconservative scheme is used in the low-Mach regime, while the conservative scheme provides sharper and more accurate results at higher Mach numbers.
Numerical tests demonstrate that the proposed method achieves second-order accuracy and that its time-step restriction is independent of the Mach number. This makes the scheme a robust and efficient alternative to standard explicit methods.
Junming Duan, University of Wurzburg, Germany
Title: Active Flux Methods for Compressible Flows: Flux Vector Splitting and Bound-Preservation
Abstract: The active flux (AF) method is a compact high-order finite volume method that simultaneously evolves cell averages and point values at cell interfaces. Within the method of lines framework, the existing Jacobian splitting-based point value update incorporates the upwind idea but suffers from a stagnation issue for nonlinear problems due to inaccurate estimation of the upwind direction, and also from a mesh alignment issue partially resulting from decoupled point value updates. I will talk about using flux vector splitting for the point value update, offering a natural and uniform remedy to the two issues. To improve robustness, I will also talk about bound-preserving (BP) AF methods for hyperbolic conservation laws, with applications to the compressible Euler equations and ideal magnetodynamics. In addition, a shock sensor-based limiting will be discussed for suppressing oscillation.
Michael Dumbser, University of Trento, Italy
Title: A Simple and General Framework for the Construction of Exactly div-curl-grad Compatible Discontinuous Galerkin Finite Element Schemes on Unstructured Simplex Meshes
Abstract: We introduce a new family of discontinuous Galerkin (DG) finite element schemes for the discretization of first order systems of hyperbolic partial differential equations (PDE) on unstructured simplex meshes in two and three space dimensions that respect the two basic vector calculus identities exactly also at the discrete level, namely that the curl of the gradient is zero and that the divergence of the curl is zero. The key ingredient here is the construction of two compatible discrete nabla operators, a primary one and a dual one, both defined on general unstructured simplex meshes in multiple space dimensions. Our new schemes extend existing cell-centered finite volume methods based on corner fluxes to arbitrary high order of accuracy in space. An important feature of our new method is the fact that only two different discrete function spaces are needed to represent the numerical solution, and the choice of the appropriate function space for each variable is related to the origin and nature of the underlying PDE. The first class of variables is discretized at the aid of a discontinuous Galerkin approach, where the numerical solution is represented via piecewise polynomials of degree N and which are allowed to jump across element interfaces. This set of variables is related to those PDE which are mere consequences of the definitions, derived from some abstract scalar and vector potentials, and for which involutions like the divergence-free or the curl-free property must hold if satisfied by the initial data. The second class of variables is discretized via classical continuous Lagrange finite elements of approximation degree M=N+1 and is related to those PDE which can be derived as the Euler-Lagrange equations of an underlying variational principle.
The primary nabla operator takes as input the data from the FEM space and returns data in the DG space, while the dual nabla operator takes as input the data from the DG space and produces output in the FEM space. The two discrete nabla operators satisfy a discrete Schwarz theorem on the symmetry of discrete second derivatives. From there, both discrete vector calculus identities follow automatically.
We apply our new family of schemes to three hyperbolic systems with involutions: the system of linear acoustics, in which the velocity field must remain curl-free and the vacuum Maxwell equations, in which the divergence of the magnetic field and of the electric field must remain zero. In our approach, only the magnetic field will remain exactly divergence free. As a third model we study the Maxwell-GLM system of Munz et al., which contains a unique mixture of curl-curl and div-grad operators and in which the magnetic field may be either curl-free or divergence-free, depending on the choice of the initial data. In all cases we prove that the proposed schemes are exactly total energy conservative and thus nonlinearly stable in the L2 norm.
Guanghui Hu, University of Macao, China
Title: A Framework of High-Order Finite Volume Solutions of Euler Equations in General Domains
Abstract: High-order methods have great potential on delivering more efficient numerical simulations in CFD, however, many factors prevent the realization of such a potential, e.g., complex domain, solution reconstruction, structure preserving, solver. In this talk, we will present a framework of high-order finite volume method for steady Euler equations, consisting of a Newton iteration for the linearization of the system and a geometric multigrid method as the linear solver. The boundary of the complex domain is approximated using NURBS technique, and a non-oscillatory k-exact method is developed for the solution reconstruction, which guarantee the high order accuracy of the solution in the complex domain. In addition, a dual-weighted-residual based h-adaptive method, as well as its realization based on a C++ library AFEPack will be introduced for the further improvement of the efficiency. One example towards the optimal design of the airfoil shape will demonstrate that the framework is a competitive one in the market.
Yongle Liu, University of Zurich, Switzerland
Title: Positivity-Preserving Well-Balanced Point-Average-Moment PolynomiAl-interpreted (PAMPA) Schemes for Shallow Water Models
Abstract: In this talk, we present recent extensions of the PAMPA (semi-discrete Active Flux) method to the shallow water equations. The PAMPA method employs a globally continuous representation of the solution variables, with degrees of freedom consisting of point values located on element edges and average values within each element. The evolution of cell averages is governed by the conservative form of the partial differential equations (PDEs), while the point values—unconstrained by local conservation—are updated through a non-conservative formulation. This flexible PAMPA framework allows for a wide range of variable choices in the non-conservative formulation, including conservative variables, primitive variables, or other suitable variable sets. We begin by introducing our first generation of well-balanced PAMPA schemes that employ different variable sets in the non-conservative formulation. We then present a recently developed well-balanced PAMPA scheme based on a global flux quadrature approach [1, 2, 3, 4, 5]. In this formulation, the discretization of the source terms is obtained from the derivative of and additional flux function computed via high order quadrature of the source term. By adopting an appropriate quadrature strategy, the scheme can exactly preserve the still water equilibrium states and also exhibits a super-convergent behavior toward moving water steady states. To ensure positivity of the water depth and suppress spurious oscillations near shocks, we blend the high-order PAMPA schemes with first-order local Lax–Friedrichs schemes specifically designed to preserve both the still water equilibrium and the positivity of water height, while also handling wet-dry interfaces. Extensive numerical experiments demonstrate the accuracy, robustness, and well-balanced properties of the proposed methods.
The talk will cover the following developments:
• Rémi Abgrall, Yongle Liu. A New Approach for Designing Well-Balanced Schemes for the Shallow Water Equations: A Combination of Conservative and Primitive Formulations. SIAM Journal on Scientific Computing, 46 (2024), A3375-A3400.
• Yongle Liu, Wasilij Barsukow. An Arbitrarily High-Order Fully Well-balanced Hybrid Finite Element-Finite Volume Method for a One-dimensional Blood Flow Model. SIAM Journal on Scientific Computing, 47 (2025), pp. A2041–A2073.
• Yongle Liu. A Well-balanced Point-Average-Moment PolynomiAl-interpreted (PAMPA) Method for Shallow Water Equations with Horizontal Temperature Gradients on Triangular Meshes. SIAM Journal on Scientific Computing, 47 (2025), pp. A3185 A3211.
• Rémi Abgrall, Yongle Liu, Mario Ricchiuto. Positivity-preserving Well-balanced PAMPA Schemes with Global Flux Quadrature for One-dimensional Shallow Water Models. Arxiv preprint, arXiv:2510.26862, 2025.
References:
[1] Y. Cheng, A. Chertock, M. Herty, A. Kurganov, T. Wu, A new approach for designing moving-water equilibria preserving schemes for the shallow water equations. J. Sci. Comput., Volume: 80, 538–554, 2019.
[2] A. Chertock, S. Cui, A. Kurganov, N. Özcan, E. Tadmor, Well-balanced schemes for the Euler equations with gravitation: conservative formulation using global fluxes. J. Comput. Phys., Volume: 358, 36–52, 2018.
[3] A. Chertock, M. Herty, N. Özcan, Well-balanced central-upwind schemes for 2 × 2 systems of balance laws. In: Theory, Numerics and Applications of Hyperbolic Problems I, volume. 236 of Springer Proceedings in Mathematics & Statistics. Springer, 345–361, 2018.
[4] A. Chertock, A. Kurganov, X. Liu, Y. Liu, T. Wu, Well-Balancing Via Flux Globalization: Applications to Shallow Water Equations with Wet/Dry Fronts J. Sci. Comput., Volume: 90, Paper No. 9, 21 pp, 2022.
[5] Y. Mantri, P. Öffner, M. Ricchiuto, Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: global flux quadrature and cell entropy correction. J. Comput. Phys., Volume: 498, p. 112673, 2024.
Maria Lukacova, University of Mainz, Germany
Title: Active Flux Method for Hyperbolic Problems using Multidimensional Evolution Operators
Abstract: We present a novel third order fully-discrete Active Flux method based on the multidimensional approximate evolution operators. The latter are derived applying the theory of bicharacteristics. An approximate evolution operator is used to evolve a numerical solution at the cell interfaces to be used in the flux evaluation. We study the influence of various approximate evolution operators on the stability and accuracy of the whole Active Flux method applied to the linear wave equation system. In particular, we discuss the influence of various third-order reconstructions in space on the stability and accuracy of the Active Flux method. We present the applications to the nonlinear Euler equations. We also discuss the recent results on rigorous convergence analysis derived for the first order variant of the method.
The results have been obtained in collaboration with S. Chu, E. Chudzik, C. Helzel, A. Porfetye, and Z. Tang.
Lorenzo Micalizzi, NC State Univerity, USA
Title: A New Semi-Discrete Finite Volume Active Flux Method for Hyperbolic Conservation Laws with Application to Multifluid Flows
Abstract: We propose a new Active Flux framework for hyperbolic systems of partial differential equations that combines conservative and primitive formulations of a given system. The two formulations are discretized on overlapping staggered meshes, where both conserved and primitive variables are represented by cell averages. The update of the conserved variables is performed via simple flux evaluations of point values reconstructed out of the primitive cell averages, while a Path-Conservative Finite Volume scheme is used for the update of the primitive variables. To avoid convergence to incorrect solutions, often observed in schemes that evolve primitive variables directly, we introduce a conservative post-processing procedure that consistently couples the two formulations, ensuring both conservation and an accurate treatment of discontinuities. The proposed method is validated through a series of one- and two-dimensional benchmark problems for the Euler equations of gas dynamics. A straightforward extension to multifluid flows will also be discussed.
Huazhong Tang, Peking University, China
Title: A High-Order Bound-Preserving Active Flux Method on Adaptive Moving Meshes
Abstract: The active flux (AF) method is a compact high-order finite volume method that simultaneously evolves cell averages and point values at cell interfaces. This talk will introduce a high-order bound-preserving active flux method on adaptive moving meshes for hyperbolic conservation laws, based on the active flux method with FVS for point value updates [Duan et al, SISC 2025]. A key is the careful discretization of mesh motion metrics, which is designed to ensure the exact preservation of free-stream states on dynamically evolving grids. Mesh adaptation is driven by the iterative solution of the Euler-Lagrange equations associated with a mesh adaptation functional. Furthermore, a general BP framework is adeptly extended to the context of adaptive moving meshes. The strategy, which combines the high-order AF scheme with a first-order Lax-Friedrichs scheme in a convex manner and incorporates an artificial viscosity to maintain the positivity of the mesh transformation Jacobians, ensures the numerical solutions remain within the physically admissible set. It is rigorously proven that the resulting method, under a suitable CFL condition, satisfies the maximum principle for scalar conservation laws and preserves the positivity of density and pressure for the compressible Euler equations. The accuracy, efficiency, and robustness of the proposed scheme are validated through numerical experiments, demonstrating its superior performance compared to computations on uniform static meshes. It is a joint work with Yixiao Tang.
Eleuterio Toro, University of Trento, Italy
Title: Wave Speeds
Abstract: Wave speeds are essential to computational hyperbolics and must be computed for every specific system. Wave speeds are needed in all explicit methods to compute a time step that satisfies a stability condition; they are needed by front-tracking methods, and by some relaxation methods. Numerical fluxes of the HLL and Rusanov type for finite volume and discontinuous Galerkin schemes need wave speeds. In computational practice such wave speeds are estimated. Rarely one would attempt to use exact wave speeds; these may be expensive, or even impossible, to compute. Estimates, however, need to bound true speeds; moreover, aiming for sharp bounds is highly desirable. Surprisingly, theoretical estimation of wave speeds has not received the attention it deserves. In fact, for the Euler equations, for example, popular wave speed estimates in current use fail to bound the true wave speeds [1], [2].
Here we first briefly review existing theoretical wave speed estimates and newly proposed ones for the Euler equations, the shallow water equations and the blood flow equations [2]. Then, in the context of Rusanov-type schemes for the linear advection equation in one and two space dimensions, we carry out a theoretical study of the consequences from errors in wave speed estimates, on the monotonicity and stability properties of the derived schemes. We show that underestimates, automatically render the schemes non-monotone, even if linearly stable in a severely restricted Courant-number interval. Overestimation, on the other hand, though preserving monotonicity will always restrict the stability range [3].
Overestimation is preferable to underestimation, for two reasons. First, schemes from overestimation are monotone, and second, their stability regions are larger than those from underestimation, for equivalent displacements from the exact speed. The findings presented in this talk may prove useful in raising awareness of the potential pitfalls of a seemingly simple practical computational task, that of providing wave speed estimates. Finally, we remark that the linear theory is seen to carry over to the non-linear scalar case, but more work is needed, especially for non-linear systems. Computational examples are shown.
[1] J. L. Guermond and B. Popov. Fast estimation of the maximum wave speed in the Riemann problem for the Euler equations. Journal of Computational Physics 321 (2), 2016.
[2] E. F. Toro, A. Siviglia and L. O. Mueller. Bounds for wave speeds in the Riemann problem. Direct theoretical estimates. Computers and Fluids, Volume 209, 104640, 2020.
[3] E. F. Toro and S. A. Tokareva. Rusanov-type schemes for hyperbolic equations: wave speed estimates, monotonicity and stability. Computers and Fluids (under review), 2025.
Title: A Simple Invariant-Domain-Preserving Framework for the PAMPA Scheme
Abstract: As an active-flux-type method, the PAMPA (Point-Average-Moment PolynomiAl-interpreted) method was introduced in [R. Abgrall, Commun. Appl. Math. Comput., 5(1):370–402, 2023] as an innovative approach that combines the conservative and nonconservative formulations of a hyperbolic system of conservation laws to evolve both cell averages and point values. In this talk, we present a simple and efficient framework for designing invariant-domain-preserving (IDP) PAMPA schemes. We begin with a rigorous analysis of the IDP property for the updated cell averages in the original PAMPA scheme, highlighting the critical roles of (i) the decomposition of cell averages and (ii) enforcing midpoint values within the invariant domain. This analysis also clarifies the difficulty of using only single-state fluxes at cell interfaces—rather than two-state numerical fluxes—to guarantee that updated cell averages remain in the invariant domain. Guided by these insights, we propose a simple IDP limiter for cell midpoint values and construct a PAMPA scheme that is provably IDP for updated cell averages without any additional post-processing limiters. This contrasts with existing bound-preserving PAMPA schemes, which typically rely on extra convex limiting to blend high-order and low-order schemes. Furthermore, inspired by the Softplus and clipped ReLU functions from machine learning, we introduce an automatic IDP reformulation of the governing equations, which yields an unconditionally limiter-free IDP scheme for evolving point values. We also develop new techniques to suppress spurious oscillations in the IDP PAMPA scheme, enabling the accurate capture of strong shocks. Numerical experiments—including tests for the linear convection equation, Burgers’ equation, the shallow water equations, the compressible Euler equations, and MHD equations—demonstrate the accuracy, robustness, and shock-capturing capabilities of the proposed IDP PAMPA framework.
Feng Xiao, Institute of Science Tokyo, Japan
Title: Construct High-Order Schemes Using Multi-Moment Constrained Flux Reconstruction Formulation
Abstract: This talk presents a general formulation for constructing high-order numerical schemes by employing multi-moment constraint conditions in the reconstruction of flux functions. The proposed formulation, termed multi-moment constrained flux reconstruction (MMC-FR), fundamentally extends the flux reconstruction (FR) approach of Huynh (2007) .
In contrast to the original FR method, which enforces only continuity constraints on the flux function at cell interfaces, the MMC-FR formulation introduces additional types of constraints, such as those on spatial derivatives or point values. This extended framework can also be interpreted as a hybrid of Lagrange and Hermite interpolations, providing a unified numerical basis that accommodates a broader class of high-order schemes.
Several representative schemes are constructed within this framework and evaluated through numerical experiments to demonstrate their accuracy and robustness.
Title: High Order Adaptive Low-Rank Method for Multi-Scale Linear Kinetic Transport Equations
Abstract: In this work, we present a new adaptive rank approximation technique for computing solutions to the high-dimensional linear kinetic transport equation. The approach we propose is based on a macro-micro decomposition of the kinetic model in which the angular domain is discretized with a tensor product quadrature rule under the discrete ordinates method. To address the challenges associated with the curse of dimensionality, the proposed low-rank method is cast in the framework of the hierarchical Tucker decomposition. The adaptive rank integrators we propose are built upon high-order discretizations for both time and space. In particular, this work considers implicit-explicit discretizations for time and finite-difference weighted-essentially non-oscillatory discretizations for space. The high-order singular value decomposition is used to perform low-rank truncation of the high-dimensional time-dependent distribution function. The methods are applied to several benchmark problems, where we compare the solution quality and measure compression achieved by the adaptive rank methods against their corresponding full-grid methods. We also demonstrate the benefits of high-order discretizations in the proposed low-rank framework.
Nan Zhang, SUSTech, China
Title: An Asymptotic Preserving Dual Formulation Finite-Volume Method for the Thermal Rotating Shallow Water Equations in All Rossby Numbers
Abstract: In this talk, we extend the AP method from Chertock et al. (AP scheme for the compressible Euler equations) to address the numerical modeling of large-scale geophysical fluid dynamics using the Thermal Rotating Shallow Water (TRSW) equations. The TRSW system models geophysical flows characterized by horizontal density/temperature variations, exhibiting multi-scale dynamics due to the coexistence of fast rotational waves and slower advective processes. To efficiently address these challenges, we combine the advantages of the primitive and conservative formulations of the TRSW system: the primitive form captures the quasi-geostrophic dynamics relevant in the low Rossby number regime, while the conservative form is employed to accurately capture shocks in the high Rossby number regime. By introducing a post-processing step, we hybridize the AP scheme for the primitive system and central-upwind scheme for the conservative system to develop a dual formulation finite-volume method valid across all Rossby numbers. Time discretization is performed using a second-order semi-implicit Runge-Kutta method, while spatial discretization utilizes a path-conservative central-upwind scheme for the nonstiff terms and second-order central differences for the stiff terms. The path-conservative formulation is crucial for maintaining accuracy when strong gradients arise during transitions between compressible and incompressible regimes. The scheme can be easily proved to be AP and requires only the solution of linear systems, ensuring both correct asymptotic behavior and ease of implementation. Numerical results confirm the effectiveness of the scheme in accurately capturing fluid dynamics across a wide range of regimes.