Schedule

The context of this work is the development of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations.

A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly.

However, a result from Gottlieb et al. (2001) gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions.

Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its $L^\infty$ stability and TVD properties.

In this work we present a novel approach based on a convex combination between a first-order TVD IMEX Euler scheme and a potentially oscillatory high-order IMEX-RK scheme.

We derive and analyse the TVD property for a scalar multi-scale equation and the resulting TVD-MOOD schemes are applied to the isentropic Euler equations.

This is joint work with Victor Michel-Dansac (Inria Strasbourg)

Many interesting applications of hyperbolic systems of equations are stiff, in the sense that restrictive CFL conditions are imposed by fields that one is not really interested in tracking accurately. A typical solution in these cases is to resort to implicit time integration, but in the field of high order accurate numerical schemes for hyperbolic equations this is made very difficult by the extreme nonlinearity of the reconstruction operators.

In this talk I will illustrate an approach, that we called Quinpi, to treat nonlinear hyperbolic equations with high order accurate implicit timestepping. It is based on a third order Central WENO reconstruction, a third order DIRK time integrator and a novel idea of time limiting. The scheme is linearized as much as possible using first order accurate predictors for Runge-Kutta stages and the final scheme contains only the nonlinearity of the flux function: in particular it can be applied by solving only linear systems for linear equations. The limiting in time is needed to control spurious oscillations arising from the fact that waves can cross more than one computational cell in each timestep.

The intrinsic diversity in both geometric and mechanical properties of blood vessels contributes to the complex fluid-structure interactions (FSI) that occur between vessel walls and blood flow, posing challenges in the numerical modeling of the human cardiovascular system. In this context, a key objective is to develop a flexible computational model that employs an accurate, efficient, and robust numerical approach.

This talk explores a highly adaptable multiscale constitutive framework designed for modeling one-dimensional blood flow. We demonstrate that diverse blood propagation phenomena can be accurately represented by appropriately selecting scaling parameters linked to different characterizations of the FSI mechanism. To solve the problem numerically, we propose to employ an Asymptotic-Preserving Implicit-Explicit Runge-Kutta Finite Volume method which ensures the consistency of the scheme with various asymptotic limits of the model without compromising its efficiency even when dealing with small scaling parameters.

For many mathematical models resulting in stiff problems, only some parts of the given equations are responsible for the stiffness. For instance, in the context of advection-diffusion problems, stiffness is usually induced by the diffusion terms and implicit-explicit (IMEX) time integrators generally discretize the advection terms explicitly and treat the diffusion terms implicitly in order to avoid the corresponding parabolic time step restrictions. Recent research has shown that specific combinations of space and time discretizations even yield unconditional stability, providing grid-independent time step restrictions only based on the advection and diffusion constants. Hereby, implicit discretization of the stiff diffusion terms additionally stabilizes the explicit discretization of the remaining advection terms. This enhanced stability property can be analytically investigated by discrete energy estimates. In this talk, we focus on the spatial discretization by upwind summation-by-parts (SBP) schemes which provide a generic framework to construct robust, structure preserving approximations of higher order and introduce an upwind mechanism similar to numerical fluxes within a discontinuous Galerkin method. In this context, unconditional stability of IMEX upwind-SBP schemes for advection-diffusion problems requires compatibility of the first and second derivative upwind SBP operators. As a second example, we consider kinetic models which describe a wide range of physical processes relevant to natural and engineering sciences where a large number of particles is involved. Collisions of particles are modeled by collision operators which usually lead to stiff problems, in particular for kinetic models in diffusive scaling. In this talk, we will consider specific linear kinetic transport equations modeling radiative transfer. Exploiting the structure-preserving properties of upwind SBP schemes, numerical efficiency can be increased by specific IMEX time integrators, where the micro-macro decomposition of the kinetic equation is suitably split into implicitly and explicity discretized terms. For IMEX upwind SBP schemes, an energy analysis again yields unconditional stability, meaning that arbitrarily large time steps may be chosen near the macroscopic limit of the kinetic model. Specifically, energy stability and error estimates for such schemes will be discussed in this talk.

The goal of this work is the development of advanced numerical approaches for hyperbolic systems with stiff relaxation. In the asymptotic limit, i.e., in the scenario of an infinitesimally small relaxation parameter, denoted as ε, these systems converge to others that may differ in type from the original. Specifically, in the asymptotic limit, a parabolic behavior may emerge, as in the case of the hyperbolic heat equation. As ε approaches zero, the relaxation term becomes very strong and highly stiff, potentially leading to spurious results in numerical simulations.

Implicit-explicit (IMEX) Runge–Kutta schemes have found extensive application in the temporal evolution of hyperbolic partial differential equations (refer to [Pareschi, Russo, J. Sci. Comput., 2005] and [Boscarino, Russo, SIAM J. Sci. Comp., 2009]). Additionally, our objective is to obtain a precisely well-balanced scheme for the asymptotic limit system. This aligns with a technique introduced by two of the authors, which relies on a well-balanced reconstruction operator (detailed in [Castro, Parés, J. Sci. Comput., 2020]). Particular emphasis is placed on accurately approximating the averages and integrals of the source terms through the use of quadrature formulas. We will present numerical experiments to assess the effectiveness of the methods and verify the well-balanced property.

Arbitrary Derivative (ADER) [Dumbser et al., JCP, 2008] and Deferred Correction (DeC) [Abgrall, JSC, 2017] are arbitrarily high-order methods developed independently in distinct contexts, yet share notable similarities. Both methods employ an iterative process, incrementing by one the order of accuracy at each step. DeC originated as an ODE solver, then used also for more complicated space-time PDE discretizations, while ADER was initially explored as a PDE solver, particularly in its DG space-time discretization, but it has been investigated also as an ODE solver.    

Explicit and IMEX versions of these methods have found widespread application, demonstrating adaptability to nonlinear problems through the linearization of stiff terms. We investigate the stability of explicit schemes, revealing a common stability region among these methods for the same order of accuracy [Han Veiga et al., AMC, 2024]. Most implicit methods exhibit A-stability [Petri et al., in prep.], while the IMEX variants are scrutinized by constraining implicit and explicit terms alternately [Minion, Comm. Math. Sci., 2003; Hundsdorfer et al., 2003].

The IMEX formulations of DeC and ADER for PDEs manifest simpler structures in the analysis of linear problems. Our exploration employs von Neumann stability analysis for advection–diffusion and advection–dispersion problems, similarly to [Tan et al. Int. J. Num. Anal. Modeling, 2021]. Hence, we can study the stability regions for some carefully chosen free parameters that depend on time step, mesh discretization and physical coefficients [Petri et al., in prep.]. Our findings establish clear upper bounds for the time step or CFL in advection–diffusion problems. In advection–dispersion cases, the stability regions are much more complicated and show that even respecting similar constraints can lead to unstable schemes.

In recent years, there has been a growing interest in modified Patankar schemes due to their ability to maintain the positivity of analytical solutions in production–destruction systems (PDS), regardless of the chosen time step size. These schemes also ensure the conservation property and have found successful applications in biological and chemical systems, as well as serving as effective time-integrators in the semi-discretization of hyperbolic balance laws. However, the stability conditions of modified Patankar schemes and their theoretical properties when applied as time-integrators in hyperbolic balance laws were not clear for a considerable period. The first part of our discussion delves into recent advancements in this field, with a focus on the numerical investigation proposed in [Torlo, Oeffner, Ranocha, App. Numer. Math., 2022], an analytical investigation based on Lyapunov stability from [Izgin, Oeffner, ESAIM: M2AN, 2023], and their interrelationship [Izgin, Oeffner, Torlo, Proceedings of HYP2022, 2022].

Moving to the second part of this presentation, we extend our exploration and showcase the highly efficient application of modified Patankar schemes as a time integration method for hyperbolic balance laws, as highlighted in [Ciallella, Micalizzi, Oeffner, Torlo, Comp. Fluids, 2022]. Towards the conclusion, we present various theoretical properties [Bender, Izgin, Oeffner, Torlo, In preparation, 2024] and initiate a discussion on stability.

This is joint work mainly with D. Torlo (Sissa) and T. Izgin (Uni Kassel). Further contributions are done with J. Bender (Uni Kassel), H. Ranocha (Johannes-Gutenberg University Mainz), M. Ciallella (University Bordeaux) and L. Micalizzi (University Zuerich).

We consider high-order Diagonally Implicit Runge-Kutta time integration methods coupled with Discontinuous Galerkin space reconstructions for conservation laws. We analyze the dispersion and diffusion properties for different pairs of schemes to select the best combination for high Courant numbers in terms of controlling spurious oscillations. Working with high-order methods, however, it is necessary to introduce local slope limiters. We therefore propose to control numerical oscillations by limiting only the high-order terms of the DG reconstruction, using appropriate weights based on a first-order predictor.

The efficient numerical solution of compressible Euler equations of gas dynamics poses several computational challenges. Severe time restrictions are required by standard explicit time discretization techniques, in particular for flow regimes characterized by low Mach number values. We propose an efficient, accurate, and robust solver for Euler equations. In addition to the results presented in previous works, we prove the asymptotic-preserving (AP) property of the numerical scheme in the low Mach number limit. The analysis is carried out considering a general equation of state (EOS) and therefore it is not restricted to the ideal gas law as done for standard AP schemes. The method is based on an Additive Runge Kutta IMEX (ARK-IMEX) method for time discretization and allows to simulate low Mach number regimes at a reduced computational cost, while maintaining full accuracy also in case of higher Mach number regimes. We couple implicitly the energy equation to the momentum one, while treating the continuity equation in an explicit fashion. The scheme has been implemented in the framework of the deal.II numerical library, using an h-adaptive Discontinuous Galerkin spatial discretization, so that adaptive mesh refinement capabilities are employed to enhance efficiency. The method is effective also for real gases equations of state such as the general cubic EOS or the stiffened gas equation of state (SG-EOS). A number of numerical experiments on classical benchmarks for compressible flows, in particular applications for atmosphere dynamics, and their extension to real gases demonstrate the properties of the proposed scheme.

Joint work with Luca Bonaventura (Politecnico di Milano).

The aim of this work is to apply a semi-implicit (SI) strategy in an implicit-explicit (IMEX) Runge–Kutta (RK) setting introduced in Boscarino et al. (J Sci Comput 68:975–1001, 2016) to a sequence of 1D time-dependent partial differential equations (PDEs) with high order spatial derivatives. This strategy gives a great flexibility to treat these equations, and allows the construction of simple linearly implicit schemes without any Newton’s iteration. Furthermore, the SI IMEX-RK schemes so designed does not need any severe time step restriction that usually one has using explicit methods for the stability, i.e. Delta t = O(\Delta t^k) for the k-th (k >= 2) order PDEs. For the space discretization, this strategy is combined with finite difference schemes. We illustrate the effectiveness of the schemes with many applications to dissipative, dispersive and biharmonic-type equations. Numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy.

Runge-Kutta (RK) methods may demonstrate order reduction when applied to stiff problems. This talk explores the issue of order reduction in Runge-Kutta methods specifically when dealing with linear and semilinear stiff problems. First, I will introduce Diagonally Implicit Runge-Kutta (DIRK) methods with high Weak Stage Order (WSO), capable of mitigating order reduction in linear problems with time-independent operators. Following that, I will discuss explicit RK methods with high WSO, tailored for the initial-boundary value problem with time-dependent boundary conditions in hyperbolic fields. On the theoretical front, I will present order barriers relating the WSO of an RK scheme to its order and the number of stages for fully-implicit RK, DIRK, and ERK schemes, serving as a foundation to construct schemes with high WSO. Lastly, I will conclude by presenting stiff order conditions for semilinear problems, essential to extend beyond the limitations of WSO, which primarily focused on linear problems.

In this talk, we present a class of higher-order [1] and high-resolution [2] compact implicit or semi-implicit methods for some representative hyperbolic partial differential equations used in conservation laws or in level set methods. Our motivation is similar to fully implicit time discretization methods for conservation laws in [3,4], that is, to offer unconditionally stable numerical schemes. Opposite to the quoted methods where the spatial and temporal discretizations are applied separately, we couple the both discretizations to obtain algebraic systems that are much easier to solve than in the case of fully implicit schemes. 

The methods can be formally derived using a partial Lax-Wendroff (or Cauchy-Kowalevski) procedure where the time derivatives in the Taylor series are replaced by the mixed derivatives exploiting the partial differential equation. This is a main modification of the standard L-W procedure in which the time derivatives are replaced by spatial derivatives only. Using some appropriate approximations of the mixed derivatives, one obtains a convenient form of the Jacobian for the resulting nonlinear algebraic equations (or the matrix in the case of linear equations) when the system of equations can be solved efficiently using, for example, a fast sweeping strategy [5].

The details will be given for nonlinear nonconservative advection equations in two-dimensional case for which unconditionally stable third order accurate method is derived [1] and for nonlinear conservation laws in one-dimensional case for which TVD scheme is derived in [2].

[1] Frolkovič, P., & Gajdošová, N. (2024). Unconditionally stable higher order semi-implicit level set method for advection equations. Applied Mathematics and Computation, 466.
[2] Frolkovič, P., & Žeravý, M. (2023). High resolution compact implicit numerical scheme for conservation laws. Applied Mathematics and Computation, 442.
[3] Arbogast, Todd, et al. (2020). A third order, implicit, finite volume, adaptive Runge–Kutta WENO scheme for advection–diffusion equations." Computer Methods in Applied Mechanics and Engineering 368.
[4] G. Puppo, M. Semplice, G. Visconti (2022). Quinpi: integrating conservation laws with CWENO implicit methods. Commun. Appl. Math. Comput.
[5] E. Lozano, T. D. Aslam (2021). Implicit fast sweeping method for hyperbolic systems of conservation laws. J. Comput. Phys., 430.