2025/12/06-2025/12/08
Yibing Chen
Title: Recent Developments of Single-Stage High Order Schemes
Abstract: In the era of Exascale computing, high-order schemes have attracted unprecedented attention from engineering applications. This report focuses on the single-stage schemes which can approach arbitrary high order accuracy in both space and time. The newly work will show how to use only compact spatial templates and without introducing additional internal degree-of-freedom equations, while allowing the use of optimal time steps. Finally, some extensions of the new scheme, including in the framework of curvilinear grids and local time stepping, will be also discussed.
Yongle Liu
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.
Acknowledgements：
Y. Liu thanks the UZH Postdoc Grant, 2024 / Verfügung Nr. FK-24-110 and SNFS grant 200020_204917.
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
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.
Jianxian Qiu
Title: A Fifth-Order Conservative Hermite WENO Remapping Method and Its Application to Moving Mesh Solver
Abstract: In this presentation, we propose a fifth-order intersection-based remapping method built on Hermite weighted essentially non-oscillatory (HWENO) reconstruction. Remapping methods are designed to conservatively interpolate the solution from one mesh onto a possibly uncorrelated new one. To obtain the information of the solution on the new mesh, our remapping method can be viewed as three basic steps: calculate the intersections between the old and new meshes exactly, approximate the solution values on the old mesh by a fifth-order positivity-preserving HWENO reconstruction, and compute the numerical integration in each overlapping region of the cell on the new mesh. The goal of this work is to show that our remapping method maintains high-order accuracy of HWENO reconstruction while preserving some desired properties, such as robustness, conservation, positivity, high resolution, and high efficiency. A series of numerical results is given to verify the efficiency and effectiveness of the HWENO remapping method on 1D lined meshes and 2D quadrilateral meshes. Furthermore, we apply the HWENO remapping method to fifth-order finite volume rezoning moving mesh HWENO (RMM-HWENO) method for hyperbolic conservation laws. Numerical results of compressible Euler equations are also presented to show the performance of the HWENO remapping method in the RMM-HWENO simulation.
This is a Joint work with Kexin Zhang and Min Zhang