arXiv subjects

Michael Dumbser

Publications and source records attributed to Michael Dumbser.

At least 19 recordsLinked to original sources

arXiv2026

Analysis and structure-preserving discretization of the Heat-GLM system

In this paper, we study a prototype hyperbolic system arising from the coupling of Generalized-Lagrangian-Multiplier (GLM) curl-cleaning with linear acoustics. We first show that, in the absence of dissipation, the equations can be rigorously derived from an underlying variational principle and that the resulting system is symmetric-hyperbolic. Further analysis shows that it not only conserves the total energy but also admits a set of invariants consisting of quadratic combinations of differential operators applied to the state variables. The addition of a stiff relaxation source term allows the system to be extended to dissipative dynamics, describing, for example, Cattaneo-type heat transfer processes that are asymptotically compatible with the Fourier law in the stiff relaxation limit; in this case, the total energy and invariants are dissipated accordingly. A new semi-implicit compatible numerical scheme on staggered grids is developed to solve this system while exactly preserving its key properties at the discrete level. In particular, we prove that the scheme is asymptotic-preserving in the Fourier limit, with a convergence rate depending on the initial data. We also show that the scheme conserves exactly the total energy as well as all the invariants in the absence of relaxation, and dissipates them consistently in its presence. These findings are demonstrated on a set of representative test cases.

math.NA↗

arXiv2026

An Arbitrary-Lagrangian-Eulerian solver for relativistic detonation waves

In this paper we study the dynamics of relativistic detonation waves theoretically and numerically. The reaction is physically accounted for by an extra term in the definition of the total energy density and by an additional equation for the evolution of the mass fraction of the reactant, while leaving formally unmodified the equations of mass and energy-momentum conservation. In this way, the Rankine-Hugoniot relations maintain the same formal structure of the inert version. For the numerical solution we use a second order finite volume ALE scheme with TVD reconstruction, where the mesh velocity is chosen equal to the shock speed. We also adopt a locally implicit algorithm for the treatment of potentially stiff reaction source terms that arise in the equation of the reactant. We furthermore propose a particularly efficient algorithm for the conversion from the conserved to the primitive variables, which for the relativistic Euler equations is known to be nontrivial. Following this approach, we can successfully solve the Zel'dovich-von Neumann-Doering profile of a relativistsic detonation wave, up to Lorentz factors of the shock front $\gamma_S\sim 7$. Our analysis allowed us to highlight a new special relativistic effect, which has remained unnoticed so far. While in Newtonian detonations the Zel'dovich pressure jump decreases monotonically with the mass flux through the shock front, in the relativistic case it shows a minimum and then rises monotonically as a function of the mass flux. This may have interesting physical implications on the amount of energy that can be extracted from a relativistic detonation wave.

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arXiv2026

On structure-preserving and pointwise conservative continuous DG schemes for hyperbolic systems

We present a new class of structure-preserving semi-discrete continuous-discontinuous Galerkin (CG-DG) finite element schemes for linear and nonlinear hyperbolic systems of partial differential equations on unstructured simplex meshes that automatically satisfy the following properties: i) the new schemes are not only cellwise conservative, but also locally pointwise conservative everywhere, hence they satisfy the integral form of the conservation law on arbitrary control volumes that do not have to coincide with the mesh at all; ii) the new methods naturally satisfy the two basic vector calculus identities $\nabla \cdot \nabla \times \mathbf{A}$ and $\nabla \times \nabla Z$ exactly pointwise locally and globally everywhere on the discrete level; iii) for linear symmetric hyperbolic systems the schemes are naturally energy conservative for the square energy, i.e. nonlinearly stable in the $L^2$ norm. The key ingredient of the new CG-DG schemes is the use of two different but compatible approximation spaces: the classical DG space $\mathcal{U}_h^N$ of discontinuous piecewise polynomials of degree up to $N$ and a classical finite element space $\mathcal{W}_h^{N+1}$ of globally continuous piecewise polynomials of degree $N+1$. In the new CG-DG schemes, the discrete solution $\mathbf{u}_h$ is sought in $\mathcal{U}_h^N$, while a suitable discrete flux field $\tilde{\mathbf{f}}_h$ is computed in $\mathcal{W}_h^{N+1}$. For $N=0$ our new schemes are directly related to cell-centered finite volume schemes with suitable vertex-based fluxes. All claimed properties of the schemes are first mathematically proven and are then also verified via suitable numerical tests. We show applications of our approach to three linear and nonlinear hyperbolic systems.

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arXiv2026

A structure-preserving semi-implicit finite volume scheme on vertex-staggered unstructured meshes

We present a novel structure-preserving semi-implicit finite volume method on vertex-based staggered meshes for the compatible discretization of first order systems of time-dependent partial differential equations (PDEs). The method preserves divergence-free and curl-free vector fields exactly thanks to the compatible vertex-staggered discretization of the state variables on unstructured grids that are constituted by primal Delaunay triangles and their dual polygons. For the weakly compressible Euler equations, the scheme is asymptotic preserving, yielding a consistent discretization of the incompressible limit as the Mach number goes to zero. The new scheme applies to a broad spectrum of PDEs, including the weakly compressible and incompressible Euler and Navier-Stokes equations, the incompressible magnetohydrodynamics (MHD) system, and the incompressible version of 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. Scalar quantities (pressure, density, viscous stress) are defined at nodes, with pressure updated implicitly in a continuous finite element fashion, yielding a symmetric and positive definite pressure system. Instead, vector fields (velocity, momentum, magnetic and distortion fields) are stored at triangle barycenters and evolved explicitly using a compatible finite volume scheme. Thanks to the semi-implicit discretization, the CFL condition is independent of the sound speed, allowing simulations at low Mach numbers. The fully compatible formulation ensures exactly divergence-free velocity field in the incompressible limit, exactly divergence-free magnetic field for MHD, and exactly curl-free inverse deformation gradient in solid mechanics. The method is validated through a wide set of test cases.

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arXiv2025

High order numerical discretizations of the Einstein-Euler equations in the Generalized Harmonic formulation

We propose two new alternative numerical schemes to solve the coupled Einstein-Euler equations in the Generalized Harmonic formulation. The first one is a finite difference (FD) Central Weighted Essentially Non-Oscillatory (CWENO) scheme on a traditional Cartesian mesh, while the second one is an ADER (Arbitrary high order Derivatives) discontinuous Galerkin (DG) scheme on 2D unstructured polygonal meshes. The latter, in particular, represents a preliminary step in view of a full 3D numerical relativity calculation on moving meshes. Both schemes are equipped with a well-balancing (WB) property, which allows to preserve the equilibrium of a priori known stationary solutions exactly at the discrete level. We validate our numerical approaches by successfully reproducing standard vacuum test cases, such as the robust stability, the linearized wave, and the gauge wave tests, as well as achieving long-term stable evolutions of stationary black holes, including Kerr black holes with extreme spin. Concerning the coupling with matter, modeled by the relativistic Euler equations, we perform some special relativistic Riemann problems, a classical test of spherical accretion onto a Schwarzschild black hole, as well as an evolution of a perturbed non-rotating neutron star, demonstrating the capability of our schemes to operate also on the full Einstein-Euler system. Altogether, these results provide a solid foundation for addressing more complex and challenging simulations of astrophysical sources through DG schemes on unstructured 3D meshes.

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arXiv2025

Structure-preserving schemes for nonlinear symmetric hyperbolic and thermodynamically compatible systems of partial differential equations

This paper aims at developing exactly energy-conservative and structure-preserving finite volume schemes for the discretisation of first-order symmetric-hyperbolic and thermodynamically compatible (SHTC) systems of partial differential equations in continuum physics. Due to their thermodynamic compatibility the class of SHTC systems satisfies an additional conservation law for the total energy and many PDE in this class of equations also satisfy stationary differential constraints (involutions). First, we propose a simple semi-discrete cell-centered HTC finite volume scheme that employs collocated grids and that is compatible with the total energy conservation law, but which does not satisfy the involutions. Second, we develop a fully discrete semi-implicit finite volume scheme that conserves total energy and which can be proven to satisfy also the involution constraints exactly at the discrete level. This method is a vertex-based staggered semi-implicit scheme that preserves the basic vector calculus identities $\nabla \cdot \nabla \times A = 0$ and $\nabla \times \nabla \phi = 0$ for any vector and scalar field, respectively, exactly at the discrete level and which is also exactly totally energy conservative. The main key ingredient of the proposed implicit scheme is the fact that it uses a discrete version of the symmetric-hyperbolic Godunov-form of the governing PDE system. This leads naturally to sequences of symmetric and positive definite linear algebraic systems to be solved inside an iterative fixed-point method used in each time step. We apply our new schemes to three different SHTC systems. In particular, we consider the equations of nonlinear acoustics, the nonlinear Maxwell equations in the absence of charges and a nonlinear version of the Maxwell-GLM system. We also show some numerical results to provide evidence of the stated properties of the proposed schemes.

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arXiv2025

A structure-preserving semi-implicit four-split scheme for continuum mechanics

We introduce a novel structure-preserving vertex-staggered semi-implicit four-split discretization of a unified first order hyperbolic formulation of continuum mechanics that is able to describe at the same time fluid and solid materials within the same mathematical model. The governing PDE system goes back to pioneering work of Godunov, Romenski, Peshkov and collaborators. Previous structure-preserving discretizations of this system allowed to respect the curl-free properties of the distortion field and the specific thermal impulse in the absence of source terms and were consistent with the low Mach number limit with respect to the adiabatic sound speed. However, the evolution of the thermal impulse and the distortion field were still discretized explicitly, thus requiring a rather severe CFL stability restriction on the time step based on the shear sound speed and the finite, but potentially large, speed of heat waves. Instead, the new four-split semi-implicit scheme presented in this paper has a material time step restriction only. For this purpose, the governing PDE system is split into four subsystems: i) a convective subsystem, which is the only one that is treated explicitly; ii) a heat subsystem, iii) a subsystem containing momentum, distortion field and specific thermal impulse; iv) a pressure subsystem. The three subsystems ii)-iv) are all discretized implicitly, hence a rather mild CFL restriction based on the velocity of the continuum is imposed. The method is asymptotically consistent with the low Mach number limit and the stiff relaxation limits. Moreover, it maintains an exactly curl-free distortion field and thermal impulse in the case of linear source terms or in their absence. The scheme is benchmarked against classical test cases verifying its theoretical properties.

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arXiv2025

A structure-preserving and thermodynamically compatible cell-centered Lagrangian finite volume scheme for continuum mechanics

In this work we present a novel structure-preserving scheme for the discretization of the Godunov-Peshkov-Romenski (GPR) model of continuum mechanics written in Lagrangian form. This model admits an extra conservation law for the total energy (first principle of thermodynamics) and satisfies the entropy inequality (second principle of thermodynamics). Furthermore, in the absence of algebraic source terms, the distortion field of the continuum and the specific thermal impulse satisfy a curl-free condition, provided the initial data are curl-free. Last but not least, the determinant of the distortion field is related to the density of the medium, i.e. the system is also endowed with a nonlinear algebraic constraint. The objective of this work is to construct and analyze a new semi-discrete thermodynamically compatible cell-centered Lagrangian finite volume scheme on moving unstructured meshes that satisfies the following structural properties of the governing PDE exactly at the discrete level: i) compatibility with the first law of thermodynamics, i.e. discrete total energy conservation; ii) compatibility with the second law of thermodynamics, i.e. discrete entropy inequality; iii) exact discrete compatibility between the density and the determinant of the distortion field; iv) exact preservation of the curl-free property of the distortion field and of the specific thermal impulse in the absence of algebraic source terms. We show that it is possible to achieve all above properties simultaneously. Unlike in existing schemes, we choose to directly discretize the entropy inequality, hence obtaining total energy conservation as a consequence of an appropriate and thermodynamically compatible discretization of all the other equations.

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arXiv2025

On general and complete multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws

In this work, we introduce a framework to design multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws on general unstructured polygonal Voronoi-like tessellations. In this framework we propose two simple but complete solvers. The first method is a direct extension of the Osher-Solomon Riemann solver to multiple space dimensions. Here, the multidimensional numerical dissipation is obtained by integrating the absolute value of the flux Jacobians over a dual triangular mesh around each node of the primal polygonal grid. The required nodal gradient is then evaluated on a local computational simplex involving the d+1 neighbors meeting at each corner. The second method is a genuinely multidimensional upwind flux. By introducing a fluctuation form of finite volume methods with corner fluxes, we show an equivalence with residual distribution schemes (RD). This naturally allows to construct genuinely multidimensional upwind corner fluxes starting from existing and well-known RD fluctuations. In particular, we explore the use of corner fluxes obtained from the so-called N scheme, i.e. the Roe's original optimal multidimensional upwind advection scheme. Both methods use the full eigenstructure of the underlying hyperbolic system and are therefore complete by construction. A simple higher order extension up to fourth order in space and time is then introduced at the aid of a CWENO reconstruction in space and an ADER approach in time, leading to a family of high order accurate fully-discrete one-step schemes based on genuinely multidimensional Riemann solvers. We present applications of our new numerical schemes to several test problems for the compressible Euler equations of gas-dyanamics. In particular, we show that the proposed schemes are at the same time carbuncle-free and preserve certain stationary shear waves exactly.

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arXiv2024

Convergence of a Hyperbolic Thermodynamically Compatible Finite Volume scheme for the Euler equations

We study the convergence of a novel family of thermodynamically compatible schemes for hyperbolic systems (HTC schemes) in the framework of dissipative weak solutions, applied to the Euler equations of compressible gas dynamics. Two key novelties of our method are i) entropy is treated as one of the main field quantities and ii) the total energy conservation is a consequence of compatible discretization and application of the Abgrall flux.

math.NA↗

arXiv2024

Variational derivation and compatible discretizations of the Maxwell-GLM system

We present a novel variational derivation of the Maxwell-GLM system, which augments the original vacuum Maxwell equations via a generalized Lagrangian multiplier approach (GLM) by adding two supplementary acoustic subsystems and which was originally introduced by Munz et al. for purely numerical purposes in order to treat the divergence constraints of the magnetic and the electric field in the vacuum Maxwell equations within general-purpose and non-structure-preserving numerical schemes for hyperbolic PDE. Among the many mathematically interesting features of the model are: i) its symmetric hyperbolicity, ii) the extra conservation law for the total energy density and, most importantly, iii) the very peculiar combination of the basic differential operators, since both, curl-curl and div-grad combinations are mixed within this kind of system. A similar mixture of Maxwell-type and acoustic-type subsystems has recently been also forwarded by Buchman et al. in the context of a reformulation of the Einstein field equations of general relativity in terms of tetrads. This motivates our interest in this class of PDE, since the system is by itself very interesting from a mathematical point of view and can therefore serve as useful prototype system for the development of new structure-preserving numerical methods. Up to now, to the best of our knowledge, there exists neither a rigorous variational derivation of this class of hyperbolic PDE systems, nor do exactly energy-conserving and asymptotic-preserving schemes exist for them. The objectives of this paper are to derive the Maxwell-GLM system from an underlying variational principle, show its consistency with Hamiltonian mechanics and special relativity, extend it to the general nonlinear case and to develop new exactly energy-conserving and asymptotic-preserving finite volume schemes for its discretization.

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arXiv2024

A first-order hyperbolic reformulation of the Cahn-Hilliard equation

In this paper we present a new first-order hyperbolic reformulation of the Cahn-Hilliard equation. The model is obtained from the combination of augmented Lagrangian techniques proposed earlier by the authors of this paper, with a classical Cattaneo-type relaxation that allows to reformulate diffusion equations as augmented first order hyperbolic systems with stiff relaxation source terms. The proposed system is proven to be hyperbolic and to admit a Lyapunov functional, in accordance with the original equations. A new numerical scheme is proposed to solve the original Cahn-Hilliard equations based on conservative semi-implicit finite differences, while the hyperbolic system was numerically solved by means of a classical second order MUSCL-Hancock-type finite volume scheme. The proposed approach is validated through a set of classical benchmarks such as spinodal decomposition, Ostwald ripening and exact stationary solutions.

math.NA↗

arXiv2024

An asymptotic-preserving and exactly mass-conservative semi-implicit scheme for weakly compressible flows based on compatible finite elements

We present a novel asymptotic-preserving semi-implicit finite element method for weakly compressible and incompressible flows based on compatible finite element spaces. The momentum is sought in an $H(\mathrm{div})$-conforming space, ensuring exact pointwise mass conservation at the discrete level. We use an explicit discontinuous Galerkin-based discretization for the convective terms, while treating the pressure and viscous terms implicitly, so that the CFL condition depends only on the fluid velocity. To handle shocks and damp spurious oscillations in the compressible regime, we incorporate an a posteriori limiter that employs artificial viscosity and is based on a discrete maximum principle. By using hybridization, the final algorithm requires solving only symmetric positive definite linear systems. As the Mach number approaches zero and the density remains constant, the method converges to an $H(\mathrm{div})$-based discretization of the incompressible Navier-Stokes equations in the vorticity-velocity-pressure formulation. Several numerical tests validate the proposed method.

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arXiv2024

High-order discontinuous Galerkin schemes with subcell finite volume limiter and adaptive mesh refinement for a monolithic first-order BSSNOK formulation of the Einstein-Euler equations

We propose a high order discontinuous Galerkin (DG) scheme with subcell finite volume (FV) limiter to solve a monolithic first--order hyperbolic BSSNOK formulation of the coupled Einstein--Euler equations. The numerical scheme runs with adaptive mesh refinement (AMR) in three space dimensions, is endowed with time-accurate local time stepping (LTS) and is able to deal with both conservative and non-conservative hyperbolic systems. The system of governing partial differential equations was shown to be strongly hyperbolic and is solved in a monolithic fashion with one numerical framework that can be simultaneously applied to both the conservative matter subsystem as well as the non-conservative subsystem for the spacetime. Since high order unlimited DG schemes are well-known to produce spurious oscillations in the presence of discontinuities and singularities, our subcell finite volume limiter is crucial for the robust discretization of shock waves arising in the matter as well as for the stable treatment of puncture black holes. We test the new method on a set of classical test problems of numerical general relativity, showing good agreement with available exact or numerical reference solutions. In particular, we perform the first long term evolution of the inspiralling merger of two puncture black holes with a high order ADER-DG scheme.

gr-qc↗

arXiv2024

A monolithic first--order BSSNOK formulation of the Einstein--Euler equations and its solution with path-conservative finite difference CWENO schemes

We present a new, monolithic first--order (both in time and space) BSSNOK formulation of the coupled Einstein--Euler equations. The entire system of hyperbolic PDEs is solved in a completely unified manner via one single numerical scheme applied to both the conservative sector of the matter part and to the first--order strictly non--conservative sector of the spacetime evolution. The coupling between matter and space-time is achieved via algebraic source terms. The numerical scheme used for the solution of the new monolithic first order formulation is a path-conservative central WENO (CWENO) finite difference scheme, with suitable insertions to account for the presence of the non--conservative terms. By solving several crucial tests of numerical general relativity, including a stable neutron star, Riemann problems in relativistic matter with shock waves and the stable long-time evolution of single and binary puncture black holes up and beyond the binary merger, we show that our new CWENO scheme, introduced two decades ago for the compressible Euler equations of gas dynamics, can be successfully applied also to numerical general relativity, solving all equations at the same time with one single numerical method. In the future the new monolithic approach proposed in this paper may become an attractive alternative to traditional methods that couple central finite difference schemes with Kreiss-Oliger dissipation for the space-time part with totally different TVD schemes for the matter evolution and which are currently the state of the art in the field.

gr-qc↗

arXiv2024

Well-balanced high order finite difference WENO schemes for a first-order Z4 formulation of the Einstein field equations

In this work we aim at developing a new class of high order accurate well-balanced finite difference (FD) Weighted Essentially Non-Oscillatory (WENO) methods for numerical general relativity, which can be applied to any first-order reduction of the Einstein field equations, even if non-conservative terms are present. We choose the first-order non-conservative Z4 formulation of the Einstein equations, which has a built-in cleaning procedure that accounts for the Einstein constraints and that has already shown its ability in keeping stationary solutions stable over long timescales. Upon the introduction of auxiliary variables, the vacuum Einstein equations in first order form constitute a ...

gr-qc↗

arXiv2024

An all Mach number semi-implicit hybrid Finite Volume/Virtual Element method for compressible viscous flows on Voronoi meshes

We present a novel high order semi-implicit hybrid finite volume/virtual element numerical scheme for the solution of compressible flows on Voronoi tessellations. The method relies on the flux splitting of the compressible Navier-Stokes equations into three sub-systems: a convective sub-system solved explicitly using a finite volume (FV) scheme, and the viscous and pressure sub-systems which are discretized implicitly at the aid of a virtual element method (VEM). Consequently, the time step restriction of the overall algorithm depends only on the mean flow velocity and not on the fast pressure waves nor on the viscous eigenvalues. As such, the proposed methodology is well suited for the solution of low Mach number flows at all Reynolds numbers. Moreover, the scheme is proven to be globally energy conserving so that shock capturing properties are retrieved in high Mach number flows. To reach high order of accuracy in time and space, an IMEX Runge-Kutta time stepping strategy is employed together with high order spatial reconstructions in terms of CWENO polynomials and virtual element space basis functions. The chosen discretization techniques allow the use of general polygonal grids, a useful tool when dealing with complex domain configurations. The new scheme is carefully validated in both the incompressible limit and the high Mach number regime through a large set of classical benchmarks for fluid dynamics, assessing robustness and accuracy.

math.NA↗

arXiv2024

A unified SHTC multiphase model of continuum mechanics

In this paper, we present a unified nonequilibrium model of continuum mechanics for compressible multiphase flows. The model, which is formulated within the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) equations, can describe the arbitrary number of phases that can be heat-conducting inviscid and viscous fluids, as well as elastoplastic solids. The phases are allowed to have different velocities, pressures, temperatures, and shear stresses, while the material interfaces are treated as diffuse interfaces with the volume fraction playing the role of the interface field. To relate our model to other multiphase approaches, we reformulate the SHTC governing equations in terms of the phase state parameters and put them in the form of Baer-Nunziato-type models. It is the Baer-Nunziato form of the SHTC equations which is then solved numerically using a robust second-order path-conservative MUSCL-Hancock finite volume method on Cartesian meshes. Due to the fact that the obtained governing equations are very challenging, we restrict our numerical examples to a simplified version of the model, focusing on the isentropic limit for three-phase mixtures. To address the stiffness properties of the relaxation source terms present in the model, the implemented scheme incorporates a semi-analytical time integration method specifically designed for the non-linear stiff source terms governing the strain relaxation. The validation process involves a wide range of benchmarks and several applications for compressible multiphase problems. Notably, results are presented for multiphase flows in all the relaxation limit cases of the model, including inviscid and viscous Newtonian fluids, as well as non-linear hyperelastic and elastoplastic solids.

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