We develop the structure-preserving high order discontinuous Galerkin methods for shallow water equations, which can preserve a general hydrostatic equilibrium state and positivity-preserving property under a suitable time step at the same time. Such equations mainly include the shallow water equations with non-flat bottom topography. By introducing well-balanced numerical fluxes and corresponding source term approximations, we established well-balanced schemes. We also discuss about the weak positivity property of the proposed schemes, and the positivity-preserving limiter can be applied to enforce the positivity-preserving property effectively. Numerical examples have been provided to demonstrate the good properties.

Lagrangian approach to both hydrostatic, non-dispersive in the short-waverange, and non-hydrostatic, dispersive rotating shallow-water magnetohydrodynamics is developed and used to analyze weakly and fully nonlinear waves described by the model. Hyperbolic structure in the non-dispersive case is displayed, and Riemann invariants are constructed. Characteristic equations are used to establish criteria of breaking and formation of shocks by magneto-gravity waves, and conditions of appearance of contact discontinuities in Alfvén waves. As in the case of non-magnetic rotating shallow water, rotation can not prevent breaking. The Lagrangian equations of the model are reduced to a single partial differential ``master'' equation, which is used to analyze the propagation of weakly nonlinear waves of both families, with or without weak rotation, and with or without weak short-wave dispersion. Corresponding modulation equations are constructed, and their main properties sketched. The same master equation is used to obtain fully nonlinear finite-amplitude wave solutions in particular cases of no short-wave dispersion, or no rotation.

Sixty years ago, Godunov introduced his method for solving the Euler equations of gas dynamics, thus creating the Godunov’s school of thought for the numerical approximation of hyperbolic equations. The building block of the original first-order Godunov upwind method is the solution of the conventional piece-wise constant data Riemann problem. The ADER methodology is a high-order, non-linear fully discrete one-step extension of Godunov’s method. The building block of an ADER scheme of order m + 1 in space and time is the generalized Riemann problem GRPm. This is a piece-wise smooth data Cauchy problem that includes source terms, with data represented by polynomials of arbitrary degree m, for example. There are by now several methods available to solve the GRPm. The ADER methodology operates in both the finite volume and DG finite elements frameworks. Here we review some key aspects of ADER and show some illustrative examples of very ambitious applications.

We develop efficient computational methods for oscillatory hyperbolic systems arising in non-adiabatic quantum dynamics involving band-crossing. Band-crossing is a quantum dynamical behavior that contributes to important physics and chemistry phenomena such as quantum tunneling, Berry connection, chemical reaction etc. In this talk, we will discuss our works in developing semiclassical methods for band-crossing in surface hopping. For such systems, which are hyperbolic but with oscillatory terms modeling non-adiabatic quantum transitions between different energy bands. We will introduce a nonlinear geometric method based "asymptotic-preserving" method that is accurate uniformly for all wave numbers, including the problem with random uncertain band gaps.

In this talk, we will present a high-order compact gas-kinetic scheme (GKS) for the two-layer shallow water equations, applied on a triangular mesh. The two-layer model introduces more intricate source terms than its single-layer counterpart. The innovative aspect of our approach is the evolution model at the cell interface, which furnishes both the numerical fluxes and the flow variables. For the well-balanced scheme development, the flow variables at the cell interfaces, which change over time, are employed to refresh the cell-averaged gradients. These evolving gradients are then incorporated into the discretization of the source terms within each control volume. Leveraging the cell-averaged flow variables and their gradients, we can reconstruct high-order initial data using compact stencils. The proposed compact high-order GKS is particularly advantageous for simulating flow dynamics across complex domains that are mapped with unstructured meshes. This work is a collaborative effort with Fengxiang Zhao and Jianping Gan.

When considering the numerical approximation of weak solutions of systems of conservation laws with source term, the satisfaction of discrete entropy inequalities is, in general, very difficult to be obtained. In the present talk, we present a suitable control of the artificial numerical viscosity in order to recover the expected discrete entropy inequalities. Moreover, the artificial viscosity control turns out to be very easy and it can be applied to any first-order finite volume scheme in a very convenient way.

The Riemann problem for the shallow water equations with a discontinuous bottom suffers from the non-uniqueness of the solution, complicating the application of the corresponding Riemann solver in numerical simulations. In this talk, we show how incorporating a specific physical hypothesis into the problem formulation can effectively resolve this ambiguity. Thus, assuming the discharge at the bottom discontinuity depends on initial conditions continuously, we prove the existence and uniqueness of the solution. Based on this, we present an algorithm for an exact Riemann solver. The talk will conclude with examples demonstrating an enhanced accuracy of the proposed solver for shallow water flows over complex topographies, including practical applications in river flow simulations, and a discussion of an approximate Riemann solver.

The nonlinear shallow water equations are used to model the free surface flow in rivers and coastal areas for which the horizontal length scale is much greater than the vertical length scale. They have wide applications in oceanic sciences and hydraulic engineering. In this talk, we study a numerical artifact of solving the shallow water equations over a discontinuous riverbed. For various first-order methods, we report that the numerical solution will form a spurious spike in the numerical momentum at the discontinuous point of the bottom. This artifact will cause the convergence to a wrong solution in many test cases. We present a convergence analysis to show that this numerical artifact is caused by the numerical viscosity imposed at the discontinuous point. Motivated by our analysis, we propose a numerical fix which works for the nontransonic problems.

We develop a flux globalization based well-balanced path-conservative central-upwind scheme for the two-layer thermal rotating shallow water (TRSW) equations, which arise in both oceanography and atmospheric sciences. There are several challenges in the development of numerical methods for the studied system. They are related to the presence of nonconservative terms modeling layer interaction and also to a quite complicated structure of steady-states solutions a good well-balanced numerical method should be able to exactly preserve. In order to treat the nonconservative product terms, we use the path-conservative technique, which is implemented within the flux globalization framework: the source and nonconservative terms are incorporated into the fluxes, which results in a quasi-conservative system, which is then numerically solved using the Riemann-problem-solver free central-upwind scheme. A well-balanced property of the resulting scheme is ensured by performing piecewise linear reconstruction for the equilibrium rather than the conservative variables, by development of special quadratures required for the flux globalization procedure, and by switching off a part of the numerical diffusion when the computed solution is near or at thermo-geostrophic equilibria. The advantages and excellent performance of the proposed scheme are demonstrated on a number of numerical examples.

One of the most common challenges faced in the context of the numerical resolution of the Shallow Water equations is the preservation of the positivity of the water height, especially in the context of high order methods. In fact, embedding such a feature in high order discretizations is well-known to result in severe CFL constraints.

In this talk, we will discuss an arbitrary high order numerical framework for the Shallow Water equations, characterized by positivity preservation of the water height without time step restrictions. The basic idea is to reinterpret a generic Finite Volume semidiscretization in space as a Production-Destruction system, which allows for the application of the unconditionally positivity-preserving modified Patankar Deferred Correction time integration, a modification of the standard Deferred Correction method for ordinary differential equations.

Several results will be shown to demonstrate the high order accuracy and the robustness of the approach, as well as its suitability for real-life applications.

We consider the numerical discretization of continuum-mechanical free boundary value problems for hyperbolic conservation laws as mathematical models in the bulk domains and, possibly, on the interface itself. The latter situation typically results from a dimensionally reduced modelling ansatz. They become attractive to circumvent the numerical resolution of a shallow domain that separates two full-dimensional bulk domains.A proper tracking of the interface is then essential and we discuss then a recently introduced moving-mesh concept for finite-volume methods.

To illustrate the complete approach we focus on two applications: the tracking of phase boundaries in compressible liquid-vapour flow and dimensionally mixed models for two-phase flow in fractured porous media. In the first case phase transition effects lead to non-standard interface dynamics. In the latter case the coupling conditions for the bulk domains involve the solution of evolution equations in the originally three-diemsional but geometrically extreme fractures which are represented as hypersurfaces.

When modeling and simulating geophysical flows, the Nonlinear Shallow-Water equations (SWE) are often a good choice as an approximation of the Navier-Stokes equations. Nevertheless, SWE do not take into account the effects associated with dispersive waves. To improve the nonlinear dispersive properties of the model, information on the vertical structure of the flow should be included. One way to do so is to consider non-hydrostatic pressure models, where the total pressure is split into a hydrostatic and a non-hydrostatic part. The advantage of non-hydrostatic models is that they present only first-order derivatives, which are easier to treat numerically.

The mathematical and numerical study of such models represents a difficult problem and usually involves the inversion of an elliptic operator at each time step , related to the non-hydrostatic pressure, when the model is numerically solved. To overcome this difficulty, we shall consider a relaxation approach that allows to approximate the original system by means of a modified hyperbolic system. Then, a high-order finite volume method will be applied to solve this relaxed system, resulting in an efficient and accurate approach for dispersive flows.

The flow resulting from a jet of water impacting a flat plate is part of common experience, yet surprisingly complex. It results in a standing hydraulic jump, which manifests as a stationary circular shock wave in the solution of the shallow water equations. At high Froude number, or in the absence of viscosity, the shock appears to be unstable and undergoes chaotic deformation in space and time. Most numerical methods struggle to accurately capture this phenomenon, for multiple reasons. The standing shock wave can easily lead to the formation of carbuncles — a numerical instability that scales with the mesh. The flow in the inner region is very shallow and fast, which can lead to loss of numerical depth positivity. Measures taken to deal with these issues may suppress the physical instability completely. I will show some of these pathologies as well as a recently-developed method that avoids them.

As a generalization of the Aw-Rascle-Zhang (ARZ) model, a multiclass homogenized traffic flow framework called adapted pressure (AP) ARZ model has been studied by Göttlich et.al. The solution to the AP ARZ model satisfies the minimum and maximum principles with respect to the Riemann invariants. However, such a property is not satisfied by most high-order numerical schemes. Additionally, numerical instabilities, potentially leading to simulation failures, arise when the numerical scheme does not sustain the non-negativity of the density and velocity. In this talk, we consider a convex invariant region induced by the aforementioned minimum and maximum principles, and design bound-preserving (BP) high-order oscillation-eliminating disconitnuous Galerkin methods for AP ARZ model. In addition to preserving discretely this invariant region and the non-negativity of traffic density and velocity, the sought BP property also helps to regulate velocity over- and undershoots in the vincinity of near-vacuum traffic states. The proposed schemes can be implemented not only on a single road segment, but also on networks. Furthermore, the proposed schemes are applicable to the degenerate cases of APARZ model, for example, the original ARZ model and the Lighthill-Whitham-Richards models. This work makes the first attempt to design and analyze high-order BP numerical schemes for ARZ or AP ARZ models. Several numerical examples are included to demonstrate the effectiveness, accuracy, and BP property of our schemes.

We develop a flux globalization based well-balanced (WB) path-conservative central-upwind (PCCU) scheme for the one-dimensional shallow water flows in channels. Challenges in developing numerical methods for the studied system are mainly related to the presence of nonconservative terms modeling the flow when the channel width and bottom topography are discontinuous. We use the path-conservative technique to treat these nonconservative product terms and implement this technique within the flux globalization framework, for which the friction and aforementioned nonconservative terms are incorporated into the global flux: This results in a quasi-conservative system, which is numerically solved using the Riemann-problem-solver-free central-upwind scheme. The WB property of the resulting scheme (that is, its ability to exactly preserve both still- and moving-water equilibria at the discrete level) is ensured by performing piecewise linear reconstruction for the equilibrium variables rather than the conservative variables, and then evaluating the global flux using the obtained point values of the equilibrium quantities. A fast and robust numerical solver is developed to convert the equilibrium variables to conservative ones. The robustness and excellent performance of the proposed flux globalization based WB PCCU scheme are demonstrated in several numerical examples with both continuous and discontinuous channel width and bottom topography. In these examples, we clearly demonstrate the advantage of the proposed scheme over its simpler counterparts.

We present a novel class of high-order Runge–Kutta (RK) discontinuous Galerkin (DG) schemes for hyperbolic conservation laws. The new method extends beyond the traditional method of lines framework and utilizes stage-dependent polynomial spaces for the spatial discretization operators. To be more specific, two different DG operators, associated with P^k and P^{k-1} piecewise polynomial spaces, are used at different RK stages. The resulting method is referred to as the sdRKDG method and features fewer floating-point operations and may achieve larger time step sizes. We have also conducted von Neumann analysis for the stability and error of the sdRKDG schemes for the linear advection equation in one dimension. Numerical tests are provided to demonstrate the performance of the new method.

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.

Rotating shallow water equations (RSWEs) are widely used to describe the dynamics of geophysical fluid systems, such as atmospheric circulation and oceanic currents. In this work, we are interested in numerical simulations of RSWEs in the quasi-geostrophic distinguished limit, where the Rossby number, which represents the ratio of the rotation time divided by the fluid flow time scale, and the Froude number, which represents the ratio of the fluid flow velocity divided by the gravity wave speed, are of the same magnitude. We employ an implicit-explicit Runge-Kutta time discretization to efficiently handle stiff and non-stiff components of the system, coupled with well-balanced finite difference weighted essentially non-oscillatory reconstructions. The new ingredients are that in order to capture the dynamics of potential vorticity in the quasi-geostrophic limit, we add the explicit time evolution of the potential vorticity, which can help to easily design an asymptotic preserving scheme leading to the correct asymptotic limit, while avoid complicate nonlinear iterative solvers. We can formally prove that the proposed scheme is well-balanced, asymptotic-preserving, as well as asymptotic-accurate. Numerical examples are provided to demonstrate the effectiveness of the proposed scheme in accurately capturing dynamics across various regimes, highlighting its advantages in large-scale geophysical fluid dynamics.

We consider weakly compressible flows coupled with cloud system that models dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a suitable finite volume method with the spectral-type approximation based on the generalized polynomial chaos expansion in the stochastic space. The finite volume method takes into account stiffness of the model due to low Mach number flows. To this goal we split the flow equations into a linear stiff and a nonlinear non-stiff subsystem and apply a globally stiffly accurate IMEX time discretization. This yields an asymptotic preserving method being uniformly stable and accurate with respect to the Mach number. The resulting numerical scheme yields a second-order accurate approximation in both space and time and exponential convergence in the stochastic space. Our numerical results demonstrate the reliability and robustness of the stochastic Galerkin method.

This talk focuses on modelling traffic flow dynamics using hyperbolic systems. After reviewing fundamental properties of traffic flow and major contributions in the field, it highlights essential research needs and numerical challenges for future directions, particularly in the context of mixed traffic flows involving connected, automated, and human-driven vehicles.

Widespread empirical observations of time series of macroscopic traffic states, described in terms of average fluid-like quantities such as flow, density, and speed across various countries on different continents, have unravelled wave-like properties of traffic flow. These include the emergence and propagation of shock waves, stop-and-go waves, and traffic instabilities. Since the seminal works of Lighthill and Whitham (1955) and Richards (1956), the use of hyperbolic systems for modelling traffic flow dynamics has been an ongoing practice. Numerous models have been proposed to capture various aspects of these dynamics. Unlike purely physical systems, traffic flow dynamics are significantly influenced by the role of human drivers and their real-time decision-making processes. Numerous physical and numerical challenges arise when incorporating such aspects into traffic flow models using hyperbolic systems. These challenges are likely to be amplified in the upcoming decades as traffic flow transitions to a mixed condition with the introduction of connected and automated vehicles, sharing the road with human-driven vehicles.

In this presentation, we introduce a novel wet-dry front reconstruction technique tailored for flows over complex geometries, encompassing variable bottom topographies and spatially dependent channel widths. This method facilitates the design of an initial data reconstruction that is both well-balanced and ensures positivity preservation, achieved through a two-step algorithm comprising marker and constructor stages. By integrating a simplified approximation of the source terms, we achieve a well-balanced property of the finite volume scheme. Additionally, the positivity-preserving property is upheld through the application of a draining time methodology. A series of numerical experiments have validated the robustness of our proposed scheme.