2026/10/17-2026/10/19
Tilte: "Construction of Entropy Satisfying Active Flux-Type Methods"
Abstract: This paper is devoted to the analysis of the entropy stability properties of Active Flux-type scheme for a hyperbolic system equipped with one entropy inequality. We also refine, with respect to [1], the consistency assumptions needed by these schemes. This type of scheme evolves two sets of degrees of freedom: point values that are chosen on the boundary of the elements that cover the computational domain, and the average of the solution in these elements. We show that the only thing to do is to get an entropy inequality for the average values, the point values degrees of freedom do not play any role. We construct a monolithic scheme which is bound preserving of [2], non oscillatory following [3], and entropy diminishing. The entropy condition is implemented in Tadmor’s framework[5], i.e. for the semi-discrete scheme only. The scheme is tested on the Kurganov-Popov-Petrova test case [4] which is known to be sensitive to the satisfaction of an entropy inequality. We show that our entropy correction is effective: if we do not activate the bound-preserving nor the non oscillatory condition, we get the correct solution with some spurious wiggles, as expected. Though the development, implementation and tests are done with the triangle version of the scheme, the same method can be used for polygonal meshes, following [2].
This is a joint work with Yongle Liu, University of Trento (Italy) and SUSTech (China)
References
[1] R. Abgrall. A combination of residual distribution and the active flux formulations or a new class of
schemes that can combine several writings of the same hyperbolic problem: application to the 1d Euler
equations. Commun. Appl. Math. Comput., 5(1):370–402, 2023.
[2] R. Abgrall, W. Boscheri, and Y. Liu. Virtual finite element and hyperbolic problems: The PAMPA
algorithm. J. Comput. Phys, 546:114521, 2026.
[3] Remi Abgrall and Yongle Liu. Robust pampa scheme in the dg formulation on unstructured triangular
meshes: Bound preservation, oscillation elimination, and boundary conditions. Journal of Computational
Physics, page 114962, 2026.
[4] Alexander Kurganov, Guergana Petrova, and Bojan Popov. Adaptive semidiscrete central-upwind
schemes for nonconvex hyperbolic conservation laws. SIAM J. Sci. Comput., 29(6):2381–2401, 2007.
[5] Eitan Tadmor. Entropy stability theory for difference approximations of nonlinear conservation laws and
related time-dependent problems. Acta Numerica, 12:451–512, 2003.
Yibing Chen
Tilte: "Recent Developments of High Order Gas Kinetic Schemes"
Abstract: In the era of Exascale computing, high-order schemes have attracted one of main issues in engineering applications. This report focuses on the single-stage gas kinetic schemes which can approach arbitrary high order accuracy in both space and time. This talk presents three parts. (i) By linearizing the equilibrium state and reducing the order of dissipation terms, the space–time transformation complexity is reduced, enabling the extension of classical GKS to arbitrary-order accuracy. (ii) Exploiting the built-in space–time information of GKS together with HWENO reconstruction, a compact spatial stencil is achieved without solving internal degree-of-freedom equations. (iii) Integrating the approximate Lax–Wendroff procedure further simplifies the space–time transformation.
Abstract: We present a new dual-formulation finite-volume (DF–FV) framework for simulating compressible multifluid flows with material interfaces. Standard single-fluid conservative finite-volume methods are known to generate spurious pressure and velocity oscillations near interfaces due to artificial mixing within computational cells, leading to nonphysical states that can contaminate the entire domain. Our approach addresses this long-standing difficulty by simultaneously solving the conservative and primitive forms of the underlying system and then applying post-processing based on material interface locations detected using the level set method. Away from interfaces, the post-processing replaces the primitive solution with the conservative one to maintain robustness, while in interface regions, the primitive solution is retained to prevent nonphysical averaging. This hybrid strategy avoids nonphysical averaging over material interfaces, significantly reduces oscillations, and preserves sharp contact discontinuities. In addition, a smoothness indicator identifies complex flow regions and enables adaptive strategies. Although the resulting scheme is not strictly conservative at the discrete level, the conservation error is controlled and decreases with mesh refinement. Numerical experiments demonstrate improved stability and more accurate interface representation compared with traditional conservative formulations, making the DF–FV approach an effective tool for high-resolution simulations of compressible multifluid flows.
Tilte: "New Adaptive High-Order Schemes for Hyperbolic Conservation Laws"
Abstract: We introduce new adaptive schemes for one- and two-dimensional hyperbolic systems of conservation laws. The proposed schemes combine a high-order quasi-linear finite-difference scheme in smooth regions with a second-order finite-volume low-dissipation central-upwind scheme in “rough” regions. To this end, we employ a smoothness indicator to automatically detect nonsmooth parts of the computed solution and further distinguish contact discontinuities from other “rough” structures. Near contact discontinuities, an overcompressive limiter is used to achieve a sharp resolution, while in the remaining “rough” regions a more dissipative limiter is employed to suppress possible spurious oscillations. In smooth regions, the high-order quasi-linear scheme is retained. This adaptive strategy avoids an excessive use of the overcompressive limiter, which may otherwise produce artificial kinks and staircase-like structures in smooth parts of the solution, while preserving a sharp resolution of shock and contact waves. The resulting adaptive schemes are tested on a number of challenging one- and two-dimensional numerical examples. The obtained numerical results clearly demonstrate the advantages of the proposed adaptive approach in terms of accuracy, robustness, and resolution of complex solution structures. This is joint work with Prof. Alexander Kurganov.
Tilte: "On Local Minimum Entropy Principle of High-Order Schemes for Relativistic Euler Equations"
Abstract: The minimum entropy principle (MEP), first established by E. Tadmor [Appl. Numer. Math. 2 (1986), pp. 211–219] for the nonrelativistic Euler system, states that the minimum of the initial specific entropy serves as a lower bound for the specific entropy at all future times. This fundamental principle provides the best-known a priori estimate for entropy in the nonrelativistic Euler equations and has been successfully incorporated into the design of stable numerical schemes (see, e.g., B. Khobalatte and B. Perthame [Math. Comp. 62 (1994), pp. 119–131]; X. Zhang and C.-W. Shu [Numer. Math. 121 (2012), pp. 545–563]). However, our understanding of entropy in the relativistic Euler equations remains far more limited than in the nonrelativistic case, owing to the nonlinear dependence of entropy on the conservative variables and the intricate mathematical structure of the relativistic Euler system.
In this talk, I will present our work establishing the MEP for the relativistic Euler equations with a broad class of general equations of state (EOSs) that satisfy relativistic causality. I will also discuss how to preserve its local form in high-order numerical schemes. At the continuous level, we identify a family of entropy pairs for the relativistic Euler equations with a general EOS and rigorously establish the strict convexity of entropy under a necessary and sufficient condition. At the numerical level, we develop a rigorous framework for designing provably entropy-preserving high-order schemes that ensure physical admissibility and preserve the MEP. Relativistic effects, coupled with the abstract formulation of a general EOS, introduce significant challenges not encountered in the nonrelativistic case or with the ideal EOS. In particular, entropy is a highly nonlinear, implicitly defined function of the conservative variables, making entropy preservation particularly difficult to enforce. To address these challenges, we establish a series of auxiliary theoretical results using technical inequalities. Another key innovation is the use of geometric quasi-linearization, which reformulates the nonlinear constraints as equivalent linear ones by introducing additional free parameters. These developments form the foundation of our entropy-preserving analysis.
We propose novel, robust, locally entropy-preserving high-order frameworks. A central challenge is accurately estimating the local minimum of specific entropy, particularly in the presence of shock waves at unknown locations. To address this, we introduce two new approaches for estimating local lower bounds of specific entropy, which prove effective for both smooth and discontinuous solutions. Numerical experiments demonstrate that our entropy-preserving methods maintain high-order accuracy while effectively suppressing spurious oscillations, outperforming existing techniques for estimating local entropy minima. Moreover, our approach is not limited to the relativistic Euler equations but can also be applied to other hydrodynamic models that admit an MEP.
Tilte: "Foliation Structures and Global Flow Dynamics of Scalar Hyperbolic Conservation Laws on Manifolds"
Abstract: A theoretical and computational framework for scalar hyperbolic conservation laws (sHCL) on two-dimensional closed regular manifolds is proposed. A geometry-compatible (GC) flux, defined through prescribed flux-directional vector fields, ensures consistency between surface divergence and the underlying geometry. It also induces a natural foliation of the manifold, reducing the 2D sHCL to a family of one-dimensional leaf-wise problems whose collective evolution determines the global flow dynamics.
The analysis is validated with a cp-WENO method combining the Closest Point Method (CPM) with conservative finite-difference WENO-Z discretization and global Lax-Friedrichs flux splitting in the Euclidean domain. The standard CPM re-extension, applied after each TVD-RK3 stage to enforce the constant-along-normals (CAN) property, can destroy discrete conservation near discontinuities, violate the Rankine--Hugoniot (RH) condition, and systematically delay shock propagation. To remedy this, we propose an spatiotemporal adaptively-penalized cp-WENO method (AP-cp-WENO) that locally weakens re-extension near detected shocks using a distance-based mask and shock sensor, while retaining full re-extension elsewhere to control normal-extension errors and boundary effects. Numerical tests for linear advection, inviscid Burgers equations with convex flux, and the Buckley--Leverett equation with non-convex flux on the sphere, torus, Dupin cyclide, and red-blood-cell surface show high-order accuracy in smooth regions, correct shock speeds, and ENO capturing of shocks, rarefactions, and nonlinear wave interactions.
Further Burgers simulations on the sphere and torus demonstrate that global flow topology is strongly shaped by geometry. On the sphere, the longest leaf acts as an asymptotic separatrix between rotational patterns. On the torus, the flux field determines whether invariant barriers separate counter-rotating flows or isolated singular points anchor nonlinear wave interactions.
This is a joint work with Prof. Wang-Bao Shan (Ocean University of China), Prof. Leevan Ling (HKBU), and Alex Shiu Lun Chu (HKBU).
Michael Dumbser
Tilte: "A Mixed Explicit ADER-DG and Zonally Implicit Space-Time DG Approach for Cartesian Cut Cell Meshes"
Abstract: We present high order accurate mixed explicit-implicit discontinuous Galerkin (DG) schemes for the solution of hyperbolic systems of conservation laws on Cartesian cut cell meshes, which are obtained by cutting embedded boundaries out of a regular Cartesian background mesh. As a result, Cartesian cut cell meshes may contain cut cells of arbitrarily small size, which leads to the well-known small cell problem, i.e. the classical CFL-type stability condition of explicit schemes would impose a severe restriction on the admissible time step length on small cut cells. To overcome this difficulty, we use high order explicit fully-discrete one-step ADER-DG schemes only on Cartesian cells away from the embedded boundaries, while the zone close to the embedded boundary that consists of cut cells and their direct face neighbors is discretized with an implicit space-time DG scheme. Both schemes are derived from the same weak formulation of the PDE in space-time and employ discontinuous space-time polynomials of equal degree $N$, namely the local space-time predictor of the ADER-DG scheme on the explicit cells and the space-time DG solution of the implicit cells. Both schemes are coupled with each other directly via the numerical flux.
The resulting mixed explicit-implicit scheme is conservative by construction, as both schemes share the same numerical flux, and its time step is determined only by the size of the Cartesian background cells, since the small cut cells and their direct neighbors are stabilized by the proposed zonally implicit space-time DG discretization.
The element-local space-time DG formulation of the explicit fully-discrete ADER-DG schemes makes the coupling to the zonally implicit space-time DG method very natural. We refer to the resulting scheme as ADER-ZIST (ADER-DG with Zonally Implicit Space-Time DG).
Curved embedded boundaries are represented by high order isoparametric sub-triangles inside each cut cell, and the nonlinear system of the implicit cells around the embedded boundaries is efficiently solved via an inexact matrix-free Newton--Krylov method with a simple block Jacobi preconditioner. For problems with shock waves, we add a suitable artificial viscosity (AV), which is discretized with a non-symmetric interior penalty Galerkin (NIPG) method controlled by an oscillation-eliminating smoothness sensor.
ADER-ZIST is benchmarked on a set of test problems for the compressible Euler equations, including an isentropic vortex around a cylinder, the Ringleb flow in a non-convex channel, a supersonic vortex in a quarter annulus, the low Mach number flow around a circular cylinder, the scattering of an acoustic pulse at five cylinders, the Sod shock tube with walls and the diffraction of a Mach 2 shock at a cylinder. The numerical results show that the scheme essentially reaches its nominal order of accuracy $N+1$ for smooth solutions and for polynomial approximation degrees up to $N=5$, on meshes with minimal volume fractions of the small cut cells down to the order of $10^{-7}$.
Elena Gaburro
Tilte: "A Unified High Order Computational Framework for Hyperbolic PDEs: from Fluid-Dynamics to Astrophysics, and into the 4D World of Hole-Like Elements"
Abstract: In this talk, I present a versatile and remarkably accurate computational framework designed for developing high order, structure preserving numerical schemes capable of solving a wide spectrum of hyperbolic PDEs, with applications spanning from fluid-dynamics to astrophysics. I focus in particular on a novel family of Arbitrary-Lagrangian-Eulerian ADER Discontinuous Galerkin methods operating on polyhedral tessellations. To maintain an optimal quality of the 3D moving mesh, I allow frequent topology changes. Here, I introduce a breakthrough approach for the PDE integration based on connecting different tessellations through 4D (3D+time) space-time control volumes, even when elements change their shape and neighbors.
Tatiana Kozubskaya
Tilte: "MPWENO Extension of EBR Schemes"
Abstract: The EBR framework introduces a vertex-centered finite volume method that employs quasi-one-dimensional variable reconstruction on edge-oriented extended stencils to enhance solution accuracy for conservation laws on unstructured mixed-element meshes. This higher accuracy stems from the reduction of the EBR approach to a high-order finite-difference method on translationally invariant meshes. For shock capturing, the method incorporates EBR-TVD, EBR-WENO, and EBR-MP techniques. The novel extension is the EBR-MPWENO scheme with the antidissipative correction.
For scale-resolving RANS-LES simulations of turbulent flows, hybrid EBR-type schemes combining high-accuracy central-difference, upwind, and shock-capturing approximations are employed. This approach minimizes numerical dissipation and achieves maximum accuracy, which is critical for resolving fine-scale vortex structures in LES regions. However, a key challenge lies in selecting spatially distributed weights for these combinations or determining sensitive switch sensors. To overcome this issue, the talk presents an original algorithmic approach governed by a single global parameter.
The numerical results for test problems and the Sonic Underexpanded Hot Jet case are presented.
Tilte: "Stability of 3rd DG/DDG Methods for Steady Scalar Conservation Laws with Discontinuities"
Abstract: It is well-known that convergence to steady state is notoriously hard when a high order (direct) discontinuous Galerkin (DG/DDG) method is applied for transonic and supersonic flows even with post-processing such as limiters and positivity preservation. The DG/DDG method discretizes the hyperbolic conservation laws in space in advance to obtain a system of first-order ordinary differential equations in time. As a result, the steady-state solution of the DG/DDG method is equivalent to the equilibrium point of this system. In this work, we analyze the stability of DG/DDG methods in the view point of dynamical systems for the scalar conservation law. We show that the steady-state solution of the 3rd-order DG/DDG method is not always stable in the presence of shock waves, and then we propose an artificial viscosity to stabilize the DG/DDG method and show that the artificial viscosity has to be order one of the mesh size to improve stability. Numerical results are given to verify theoretical analysis.
Tilte: "Do We Have a Crisis in Mathematical Fluid Dynamics?"
Abstract: Recent developments in mathematical fluid dynamics have revealed fundamental limitations of standard solution concepts. Finite-time blow-up of classical solutions has been established for several compressible and incompressible fluid models, while advances in convex integration have shown that the Euler system may be ill-posed in the class of weak entropy solutions: the same initial data can yield infinitely many solutions. These results indicate that the classical frameworks of strong and weak solutions alone may be insufficient.
In this talk, we discuss a new approach to recovering well-posedness for the compressible Euler equations by enlarging the solution framework to generalized solutions, specifically dissipative measure-valued solutions, and complementing this framework with suitable selection principles.
We introduce several such selection criteria, formulated as convex optimization problems subject to linear constraints arising from the dissipative measure-valued formulation. We further investigate how higher-order reconstructions and numerical limiters influence the resulting approximate measure-valued solutions and the values of the corresponding selection functionals. Numerical experiments illustrate the proposed selection principles and support the theoretical findings.
Tilte: "A High-Resolution Discontinuous Galerkin Scheme with Subcell Limiters for Multi-Material Flows"
Abstract: In this presentation, a high-order discontinuous Galerkin (DG) method is designed for volume-fraction models of compressible multi-material flows. To achieve high-resolution discontinuity capturing, a subcell limiter is incorporated, which suppresses spurious oscillations while introducing minimal numerical dissipation through a high-order weighted essentially non-oscillatory (WENO) evolution on the subcells. By combining a quasi-conservative formulation with appropriate numerical fluxes and subcell WENO reconstructions, the proposed scheme preserves the Abgrall condition and maintains equilibrium states with constant volume fractions. In addition, a bound-preserving limiter is applied, and the admissibility of the updated cell averages is rigorously established using the geometric quasi-linearization approach. A comprehensive set of two- and three-dimensional benchmark tests demonstrates the accuracy and robustness of the method.
Tilte: "High-Order Accurate Structure-Preserving Adaptive Moving Mesh Methods for 2D Compressible Euler Equations"
Abstract: Developing high-order accurate structure-preserving numerical methods for hyperbolic conservation laws is an interesting topic nowadays. We present three classes of high-order accurate structure-preserving moving mesh methods for two-dimensional compressible Euler equations on rectangular meshes and demonstrates and compares their efficiency and capability to capture discontinuities. These methods are high-order accurate entropy stable finite difference methods with multi-resolution WENO-based dissipations, positivity-preserving finite volume methods, and oscillation-eliminating nodal discontinuous Galerkin methods (OEDG) with a positivity-preserving limiter. They are built on the Euler equations in curvilinear coordinates and suitable metric discretizations, where the geometric conservation laws are essential for maintaining accuracy and preserving free-stream solutions and structures on moving meshes. The mesh points are iteratively redistributed by using the adaptive moving mesh strategy according to the monitor functions based on fluid variables, allowing the adaptive moving mesh methods to better resolve localized structures such as the shock waves and the other discontinuities etc. Several numerical experiments are conducted by using the seventh-order accurate ES finite difference method, the fifth-order accurate finite volume method, and the fourth-order accurate OEDG method to demonstrate that these three methods can achieve high resolution with fewer mesh cells and improve efficiency compared with uniform mesh methods. It is a joint work with Mrs. Yixiao Tang, Zexuan Yang, Tengfei Zheng.
Tilte: "Multi-Dimensional Equilibria Preserving Methods for Hyperbolic Equations"
Abstract: Many conservation and balance laws admit families of moving steady states that are crucial to preserve at the numerical level. While several techniques have been developed to achieve this in one dimension, few are available for multidimensional problems, and those often apply only to linear cases. The so-called Global Flux (GF) approach [Chertock, et al. 2018] integrates multiple terms into a single physical flux, yielding a unified differential operator acting on a more complex flux function. We extend this to multiple dimensions [Barsukow, et al. 2025, Barsukow, et al. 2026] both in finite element and finite volume formulations, in order to discretely preserve truly multi-dimensional equilibria as divergence free solutions. While natively defined on Cartesian grids, we also present an extension to curvilinear domains using a ghost-point boundary extrapolation method.
Tilte: "Spatially Adaptive-Order Bound-Preserving Velocity-Consistent AWENO Schemes for Compressible Two-Medium Flows"
Abstract: In this talk, a spatially adaptive-order bound-preserving (AO-BP) method for high-order characteristic-wise velocity-consistent AWENO scheme is proposed for solving the five-equation model of compressible two-medium flows with stiffened-gas equations of state. Classical bound-preserving methods with limiters (BP-Limiter) enforce physical admissibility by blending high-order and first-order fluxes, but they require a restrictive global CFL condition. The proposed AO-BP method detects inadmissible cells after each Runge-Kutta stage and locally lowers the spatial order from seventh to fifth, third, and, if necessary, first order. A conservative lowest-order interface-flux selection provides a unique numerical flux between neighboring cells using different orders. The method retains the velocity-consistent formulation, preserving pressure-velocity equilibrium at material interfaces, and uses the first-order VC scheme as an admissible fallback under the corresponding CFL condition. Numerical tests involving shock-interface interactions, near-vacuum rarefactions, gas-liquid shock tubes, underwater explosions, triple-point flows, and Mach 6 shock-water-cylinder interaction show that the method maintains positivity and volume-fraction bounds, captures discontinuities without spurious oscillations, and preserves high-order accuracy in smooth regions. In the reported two-dimensional tests, AO-BP runs successfully with CFL = 0.9 and achieves approximately twofold to threefold speedup over the classical BP-Limiter.
Tilte: "Adaptive Moving-Coordinate Optimize-then-Discretize Method for Solving Conservation Laws"
Abstract: Standard optimize-then-discretize methods are effective for smooth low-dimensional solution manifolds, but often struggle with hyperbolic conservation laws once steep gradients or shocks develop. In such regimes, global linear or smooth parameterizations may lead to ill-conditioned velocity systems, Gibbs-type oscillations near discontinuities, and a large number of basis functions to resolve moving shocks. To address these issues, we propose an adaptive moving-coordinate optimize-then-discretize (MC-OtD) method for one-dimensional hyperbolic conservation laws involving shocks, contacts, rarefactions, and steep gradients. The solution is represented by a shock-aware multi-branch ansatz in which each branch evolves in its own moving coordinate system, combined with entropy-guided topology adaptation. The branch, coordinate-map, and interface velocities are determined simultaneously from a coupled least-squares system. Numerical experiments on linear transport, Burgers-type equations, Buckley-Leverett flow, shallow-water equations, and Euler Riemann problems demonstrate that the proposed method tracks moving shocks, rarefaction fans, shock interactions, and topology changes while maintaining compact branch-wise representations and accurate conservative dynamics.
Tilte: "Well-Balanced Path-Conservative Discontinuous Galerkin Method with Generalized Hydrostatic Reconstruction for One-Dimensional Non-Conservative Balance Laws"
Abstract: This paper develops a high-order well-balanced path-conservative discontinuous Galerkin (PCDG) method for one-dimensional balance laws with possible non-conservative products, referred to as the PCDG-GHR method. Its main novelty is a coupled interface-volume construction based on generalized hydrostatic reconstruction (GHR). At interfaces, GHR uses equilibrium variables, namely variables that remain constant along the prescribed equilibrium state, to reconstruct the interface states and balance the numerical fluxes with the Dal Maso--LeFloch--Murat (DLM) path integrals. Within each cell, a reference equilibrium decomposition removes the discrete imbalance caused by projecting an exact equilibrium into the discontinuous Galerkin (DG) polynomial space. The resulting scheme exactly preserves prescribed rest and moving equilibria at the semi-discrete and SSP-RK fully discrete levels under the stated conditions. The construction is local and uses explicit time integration. Standard TVB and positivity-preserving scaling limiters remain applicable under the stated compatibility conditions. Numerical experiments for the two-layer shallow water equations and the blood flow equations confirm high-order accuracy, round-off-level equilibrium preservation, resolution of small perturbations, and robustness for discontinuous flows.