In this talk, we present a new conservative semi-Lagrangian finite difference scheme for nonlinear advection-diffusion problems. Conservation, which is fundamental for obtaining physically consistent solutions, is ensured by integrating the governing equations over a space-time control volume constructed along the characteristic curves originating from each computational point. The subsequent application of Gauss theorem allows the evaluation of all space-time surface integrals and enables the computation of the time integrals of the flux using only from spatial information available at the beginning of the timestep. For nonlinear problems, a nonlinear equation must be solved to find the foot of the characteristic, whereas this is not needed in linear cases. Furthermore, we show how the diffusion terms can be directly incorporated within the same framework, leading to the development of a novel characteristic-based Crank-Nicolson discretization in which the diffusion contribution is implicitly evaluated at the foot of the characteristic. A broad set of benchmark tests will be presented to access the accuracy, robustness, and strict conservation property of the proposed method, as well as its unconditional stability, which is verified by numerical experiments with CFL numbers up to 100.

This is a joint work with Walter Boscheri, Matteo Semplice and Maurizio Tavelli.

We consider the one-dimensional (1D) isentropic two-phase flow model presented in [Romenski, Drikakis, Toro (2010)].
This type of model in an explicit finite volume framework, requires a CFL condition that takes into account the eigenstructure of both phases, indipendently of the related volume fractions.

We introduce a new first order, splitting-based, semi-implicit numerical scheme for the 1D isentropic two-phase flow equations. This framework separates the terms related to the two phases, requiring a CFL condition based only on one of them. We choose the phase that presents the higher Mach number, consequentially the final time step is considerably higher with respect to the one required by an explicit scheme. The final numerical scheme is systematically assessed on a carefully designed suite of test problems compared with classic existing mainstream methods. A stiffness study of the source term is finally carried out.

This is a joint work with Michael Dumbser and Gabriella Puppo.

We present a novel semi-implicit hybrid finite volumes/finite elements method for the discretization of first order hyperbolic partial differential equations (PDEs). The method is fully compatible and asymptotic preserving due to the staggered discretization of conserved variables on grids constituted by Delaunay triangles and their dual star polygons.

Our method applies to a broad spectrum of governing equations, including the compressible and incompressible Euler and Navier-Stokes equations, the magnetohydrodynamics (MHD) system, and the first-order hyperbolic Godunov-Peshkov-Romenski (GPR) model for continuum mechanics.

The computational domain is covered by a primal triangular mesh and a dual tessellation made of so-called star polygons constructed by linking the barycenters of adjacent triangles and the midpoints of shared edges.

The scalar pressure field, the density, and the viscous stress tensor are defined at the triangles nodes. In particular, the pressure equation is evolved implicitly in a finite element fashion, yielding a symmetric and positive definite system. Instead, the velocity, the momentum, the magnetic and distortion fields are stored at the triangles barycenters and evolved explicitly with a finite volume approach.

Thanks to this semi-implicit conservative treatment, the CFL condition depends only on convective terms, improving computational efficiency and enabling simulations at all Mach numbers.

The fully compatible nature of the method ensures exact divergence-free velocity and magnetic field, exact curl-free for the distortion field, energy stability and correct asymptotic behavior in the incompressible limit.

Finally, extensive validation through classical benchmark test cases, including the Taylor-Green vortex, MHD vortex and the solid rotor problem, demonstrates the scheme’s accuracy, reliability, and advantages over standard numerical methods.

Deferred Correction (DeC) and Arbitrary DERivative (ADER) methods are iterative solvers used to approximate solutions of hyperbolic partial differential equations (PDEs). When the spatial discretization—typically a Discontinuous Galerkin (DG) method in the case of ADER—is factored out, these schemes can be interpreted as Runge–Kutta (RK) methods. In particular, they can be expressed in terms of two operators: a low-order RK scheme (which may be explicit, implicit, or IMEX) and a high-order implicit RK scheme, whose combination through successive iterations yields the final method. We observe that the high-order implicit RK scheme usually possesses favorable stability properties (A-stability, L-stability), and these properties are often inherited by the implicit iterative method.

I will present a structure preserving scheme to mimic at the discrete level some mathematical properties the multiphase diffuse-interface model for an arbitrary number of constituents that we have recently formulated in the framework of Hyperbolic Thermodynamically Compatible (HTC) equations. The thermodynamic compatibility at the discrete level is achieved through the numerical flux correction recently introduced by Abgrall et al. The consistency with the second law of thermodynamics is satisfied through an appropriate thermodynamically compatible parabolic vanishing viscosity regularization, as well as through an opportune discretization of the dissipative processes defined at the continuous level. In addition, space will be devoted to the derivation of an appropriate discretization that preserves the non-linear algebraic constraints on density and volume fraction physical bounds. To this end, a more classical but equivalent form of the PDEs system will be considered, which favours the time evolution of non-dimensional quantities, such as volume fractions. In this case, a simple positivity condition can be derived for the numerical solution of the non-conservative volume fraction evolution equations. This condition is then extended to the solution of the equivalent dimensional ones by satisfying a discrete product rule in the context of HTC numerical schemes. This new approach allows for a better understanding of the construction of compatible numerical fluxes in close relation to the PDE system to be solved, which is why it is also to be considered as groundwork for possible future extensions of the scheme to higher order. The validation process includes a range of benchmarks and applications to compressible multiphase problems.

This is a joint work with I. Peshkov and M. Dumbser.

In this talk, we investigate the behaviour of discontinuous Galerkin schemes for hyperbolic conservation laws, focusing particularly on the interaction between spatial discretisation and implicit time integration. The first part of the talk involves analysing how different implicit Runge–Kutta methods influence numerical diffusion and dispersion, with the aim of identifying optimal space–time discretisation combinations for high Courant numbers. In the second part, we apply this analysis to stiff hyperbolic systems. To mitigate the nonlinear complexity introduced by high-order spatial limiters, we propose precomputing these limiters using a first-order solution prediction. Numerical experiments on both scalar and system cases are presented.

Motivated by the numerical integration of stiff conservation laws, the Quinpi approach as been developed by Puppo, Semplice and Visconti. In this talk I will present the two dimensional extension of Quinpi scheme. Similarly to the one-dimensional case, the scheme uses a third order CWENO reconstruction in space combined with a third order DIRK method for time integration and a low order predictor to ease the computation of the Runge-Kutta stages. Even applying space-limiting, spurious oscillations may still appear in implicit integration, especially for large time steps. For this reason, a time-limiting procedure based on numerical entropy production is applied. I will present test cases for the Euler equations also in low Mach regimes.

This is a joint work with Matteo Semplice.

In this work, we investigate travelling wave solutions for the two-phase model proposed by Romenski et al. (2007) that describes two-phase two-fluid flow with different pressures, velocities, and temperatures. We will focus on the study of the one dimensional problem, where we examine the Rankine–Hugoniot conditions in different physical configurations to characterize admissible shock solutions. We will then focus our attention on the high-order one-step Arbitrary-Lagrangian-Eulerian (ALE) WENO finite volume scheme for the solution of nonlinear systems of hyperbolic balance laws with stiff source terms as presented by Dumbser et al. in (2015). Lastly we will show the shock profiles obtained with the ALE scheme in different regimes.

Starting from the multi-species Boltzmann equation for a gas mixture, we propose the formal derivation of the isentropic two-phase flow model introduced in [Romenski, E., and Toro, E. F., Comput. Fluid Dyn. J., 13 (2004)].  We examine the asymptotic limit as the Knudsen numbers approach zero, in a regime characterized by resonant intra-species collisions, where interactions between particles of the same species dominate. This specific regime leads to a multi-velocity and multi-pressure hydrodynamic model, enabling the explicit computation of the coefficients for the two-phase macroscopic model. The derivation also includes a treatment of the interface separating the two phases, which naturally arises from the spatial segregation of the species. In addition, our formulation also accounts for the inclusion of the evolution of the volume fraction, which is a key variable in many macroscopic multiphase models.
The derivation of the macroscopic model from the kinetic description provides several numerical insights for the development of efficient computational schemes.

This is a joint work with Gabriella Puppo and Thomas Rey.

Based on our recent progresses in the numerical solution of the Einstein equations written as a first order system in the classical 3+1 formulation, I will describe preliminary steps to derive the Einstein equations as a rigorous Euler-Lagrange system. Having the goal to obtain a non-linearly stable scheme for numerical relativity, I will review the attempts that could make this result possible, starting from Einstein choice, following Landau arguments and highlighting the main difficulties. 

I will present a compatible semi-implicit discretization for the Maxwell-GLM system, which augments the original vacuum Maxwell equations via a generalized Lagrangian multiplier approach (GLM) by adding two supplementary acoustic subsystems. The mathematical structure of the augmented Maxwell-GLM system is very intriguing and contains an interesting combination of curl-curl and div-grad operators, which is usually not present in classical PDE systems of continuum physics. We have shown that it can be derived from an underlying variational principle, has a symmetric hyperbolic structure and admits an extra conservation law for the total energy density. Therefore, the Maxwell-GLM system extends the class of symmetric hyperbolic and thermodynamically compatible (SHTC) systems established by Godunov and Romenski. I will introduce a new exactly energy-conserving and asymptotic-preserving nite volume scheme for the Maxwell-GLM system. The method introduced is a vertex-based staggered semi-implicit scheme that preserves the fundamental vector calculus identities ∇ ⋅ ∇ × A = 0 and ∇ × ∇ Φ = 0 exactly on the discrete level thanks to the use of mimetic discrete differential operators and an appropriate location of the variables on the grid. Furthermore, the structure-preserving staggered semi-implicit scheme is exactly compatible with the divergence-free condition of the electric and magnetic elds and is asymptotic-preserving. The last property of the presented scheme is that it exactly preserves the global discrete total energy in the case of periodic boundaries.