Numerical Methods for Hyperbolic Problems (numhyp25) conference with several contributions

Marco Artiano, Arpit Babbar, Louis Petri, and I will be at the conference Numerical Methods for Hyperbolic Problems (numhyp25) in Darmstadt.

Marco Artianos’s poster

Entropy-Stable High-Order Methods for the Compressible Euler Equations in Potential Temperature Formulation for Atmospheric Flows

In the last decade, there has been a growing interest in developing and studying new dynamical cores for climate and weather simulations based on the Discontinuous Galerkin (DG) approach. Various numerical methods and formulations of the Euler equations are employed to achieve accuracy, efficiency, and stability. In this work, we develop novel structure-preserving methods for the Euler equations, treating the potential temperature as a prognostic variable. We construct three numerical fluxes, ensuring entropy and total energy conservation, and extend the analysis to include a geopotential source term within the DG framework on general curvilinear meshes. Furthermore, we investigate a generalization of the kinetic energy-preserving and total energy-conserving properties when coupled with a generic geopotential term, along with well-balanced schemes for different constant background states. To achieve these properties, we adopt a flux-differencing approach for the discretization of the source term. Finally, we present several test cases to assess the theoretical findings and compare the potential temperature formulation to the traditional Euler equations formulation, on different classical atmopsheric test cases.

Arpit Babbar’s poster

Admissibility preserving IMEX compact Runge-Kutta Flux Reconstruction methods

Compact Runge-Kutta (cRK) Discontinuous Galerkin (DG) methods, recently introduced in [Q. Chen, Z. Sun, and Y. Xing, SIAM J. Sci. Comput., 46: A1327–A1351, 2024], are a variant of RKDG methods for hyperbolic conservation laws and are characterized by their compact stencil including only immediate neighboring finite elements. There are several phenomenon like chemical reactions which are modeled as stiff source terms in hyperbolic conservation laws. Explicit methods are known to fail at resolving such stiff source terms. In this talk, we present a cRK Flux Reconstruction (cRKFR) scheme with an IMplicit EXplicit (IMEX) discretization to treat stiff source terms. The implicitness is only local in space and thus does not increase the interelement communication. Inclusion of source terms in the cRKFR scheme is done by treating them as time averages. A time average source limiter is proposed that ensures admissibility in means of the IMEX scheme. The capability of the scheme to handle stiff terms is shown through tests involving Burgers’ equations, reactive Euler’s equations. The scheme is also extended to handle hyperbolic equations with non-conservative products. The extension to non-conservative products is validated through the shear shallow water equations with experimental data.

Louis Petri’s poster

Fully-discrete stability analysis of the domain of dependence stabilization for cut-cell meshes

Cut-cell meshes avoid the expensive procedure of body-fitted meshes for complex geometries. However, several issues arise with this approach, like the small-cell problem. Dealing with these is an active field of research for hyperbolic differential equations. The recent domain of dependence stabilization for discontinuous Galerkin methods addresses this problem by redistributing mass in between the neighborhood of a small cut cell on a semi-discrete level. It shows some promising results in experiments and was proven to possess a stable semi-discretization for linear equations. Our aim is to extend this to a fully-discrete stability analysis. In this talk, we present the framework and techniques to obtain strong stability by applying explicit Runge-Kutta methods and elaborate on further gained insights of the stability mechanisms that affect the time step restriction.

My talk

Recent progress on structure-preserving numerical methods for dispersive wave equations

Water wave propagation problems can often be modeled using a depth-averaged shallow water approximation, e.g., tsunami propagation or dam breaks. The classical first-order hyperbolic shallow water equations are sufficient to describe the wave dynamics in many cases. However, some applications require more accurate models, e.g., nonlinear dispersive wave equations taking higher-order effects into account. Several variants of such models exist and are used in practice, e.g., the BBM-BBM equations, the Serre-Green-Naghdi system, and hyperbolic approximations thereof. In this talk, we will review some recent developments of structure-preserving numerical methods. In particular, we will consider invariants such as the total energy and study efficient numerical methods yielding qualitative and quantitative improvements compared to standard schemes. To develop structure-preserving schemes, we make use of the general framework of summation-by-parts (SBP) operators in space, unifying the analysis of finite difference, finite volume, finite element, discontinuous Galerkin, and spectral methods. Finally, we combine structure-preserving spatial discretizations with relaxation methods in time to obtain fully-discrete, energy-conservative schemes. This talk is based on the joint works https://arxiv.org/abs/2402.16669 and https://arxiv.org/abs/2408.02665 with Joshua Lampert and Mario Ricchiuto.