arXiv subjects

Christian Rohde

Publications and source records attributed to Christian Rohde.

At least 19 recordsLinked to original sources

arXiv2026

Energy consistent hyperbolic approximation for a class of fourth-order partial differential equations

In this article, we propose a novel hyperbolic relaxation system for a general class of fourth-order nonlinear partial differential equations arising in the modelling of thin film flows or the phase separation of binary mixtures. The approximations are constructed to dissipate energies which recover the energy (Lyapunov) functional of the limit equation when the relaxation parameters vanish. Using the relative energy framework, we prove the convergence of weak entropy solutions of the relaxation system to smooth solutions of the limit equation. We validate our analysis with a series of numerical examples for thin film equations and Cahn-Hilliard equations.

math.AP↗

arXiv2026

Dimension Reduction and Asymptotic Approximation of Reactive Transport in a Bulk Domain with a Branched Thin Fracture

We study nonlinear reactive transport in a two-dimensional bulk domain containing a thin branched fracture composed of three narrow branches of thickness epsilon connected through a junction node of diameter O(epsilon). The microscopic model couples nonlinear parabolic reaction-diffusion equations in the bulk with an advection-diffusion equation in the fracture, where the longitudinal Peclet number is of order epsilon^{-1}, leading to advection-dominated transport along the branches. Nonlinear side-dependent flux conditions describe the coupling between the bulk and the fracture. As epsilon tends to zero, the fracture collapses to a one-dimensional graph, and we derive a recurrent structure of effective limit problems: a first-order hyperbolic problem on the graph satisfying the classical Kirchhoff transmission condition at the node, and reaction-diffusion equations in the bulk with nonlinear Robin conditions on the graph edges involving the graph solution. The node boundary conditions of the microscopic model do not affect these leading-order limits. To capture the influence of the node geometry, we construct node-layer and corner-layer correctors and determine subsequent terms of the asymptotic expansion. Their coefficients solve auxiliary boundary-value problems in unbounded domains with outlets at infinity and in corner-type geometries. We assemble a complete multiscale approximation combining bulk, branch, node-layer, and corner-layer contributions, and establish quantitative error estimates in appropriate energy norms. These estimates demonstrate the accuracy of the approximation relative to epsilon and depend explicitly on the corner angles of the limiting graph, reflecting the geometric complexity of the fracture network.

math.AP↗

arXiv2026

h-Adaptive FV Subcell Shock-Capturing for DGSEM on Heterogeneous Curvilinear Meshes

High-order methods offer superior dispersion and dissipation properties compared to low-order schemes but require robust stabilization for discontinuities. To ensure stability, local artificial viscosity is common, but often degrades sub-element resolution. Conversely, subcell resolution preserving limiting strategies such as the finite volume subcell method are typically restricted to uniform topologies, such as purely hexahedral or simplex meshes, or linear elements. This leaves a significant gap in treating the hybrid-element topologies necessary for complex engineering geometries. To bridge this gap, we introduce a robust shock-capturing approach for the discontinuous Galerkin spectral element method on mixed curvilinear meshes containing hexahedral, prismatic, tetrahedral, and pyramid elements. Non-hexahedral elements are handled via collapsed coordinate transformations. The proposed method utilizes an $h$-adaptive finite volume subcell scheme with an arbitrary subcell resolution up to $2\mathcal{N}+1$. Special care is taken to ensure conforming subcell distributions across different element types. Crucially, discrete conservation is proven theoretically and verified numerically, alongside validations for high-order convergence and robust shock capturing. Finally, the method's applicability to complex configurations is demonstrated through a simulation of the flow around a NACA 0012 airfoil.

math.NA↗

arXiv2026

Global Existence for a Class of Keyfitz--Kranzer Systems with Application to Thin-Film Flows

We prove the existence of global weak entropy solutions for a class of non-symmetric Keyfitz-Kranzer type systems that includes lubrication models for thin-film flow. We identify a family of entropy/entropy-flux pairs for these first-order systems, which is, in particular, admissible for a tailored second-order approximate system. The latter is motivated by higher-order dissipation operators in thin-film flow models. By identifying an invariant region in the state space, it is possible to derive a-priori $L^\infty$-bounds for the sequence of solutions to the approximate system. Exploiting the parabolic and transport structure of the equations associated with the Riemann invariants, we then rigorously justify the vanishing-diffusion limit and establish the existence of weak entropy solutions for the Cauchy problem for the first-order systems.

math.AP↗

arXiv2026

A data-driven approach for 2D vorticity PDF equations by a new conditional average estimation

We consider the statistics for the vorticity field in two-dimensional homogeneous isotropic turbulence (HIT). First, we exploit the invariance properties to derive dimensionally reduced governing equations for the one-point and two-point probability density functions (PDFs). These take the form of linear kinetic transport equations, but with an unclosed operator in terms of a conditional average. To solve the PDF equation numerically we suggest a hybrid data-driven method that relies on carefully selected samples of DNS data and a sampling estimator for the conditional average. The method is applied to DNS data for both decaying and forced HIT, demonstrating good agreement with the direct evaluation of the PDFs using the DNS data.

physics.flu-dyn↗

arXiv2026

Multiscale Hyperbolic-Parabolic Models for Nonlinear Reactive Transport in Heterogeneously Fractured Porous Media

We study nonlinear reactive transport in a layered porous medium separated by an $\varepsilon$-thin, highly heterogeneous fracture whose aperture and obstacle pattern vary periodically. Species transport in the bulk is governed by parabolic reaction--diffusion equations, coupled to a convection-diffusion-reaction problem in the fracture with nonlinear wall and obstacle reactions and Peclet number of order $O(\varepsilon^{-1})$. Via multiscale analysis as $\varepsilon \to 0$, when the fracture collapses to a flat interface, we derive a new type of homogenized model consisting of bulk diffusion--reaction equations coupled through nonlinear interface conditions and a first-order semilinear hyperbolic system on the interface. We prove well-posedness and regularity of the limit system, construct a multiscale approximation with boundary-layer correctors, and derive quantitative error estimates in suitable energy norms.

math.AP↗

arXiv2026

Convergence of a two-parameter hyperbolic relaxation system toward the incompressible Navier-Stokes equations

We investigate a two-parameter hyperbolic relaxation approximation to the incompressible Navier-Stokes equations, incorporating a first-order relaxation and the artificial compressibility method. With vanishingly small perturbations of initial velocity, we rigorously prove the simultaneous convergence of fluid velocity and pressure toward the Navier-Stokes limit in the three-dimensional case by constructing an intermediate affine system to obtain the necessary error estimates for the pressure. Furthermore, we extend the velocity convergence analysis to the case of $\mathcal O(1)$ initial velocity perturbations, and establish the global-in-time recovery of the velocity field using a modulated energy structure and delicate bootstrap arguments in both two- and three-dimensional settings.

math.AP↗

arXiv2026

Justification of a Relaxation Approximation for the Navier-Stokes-Cahn-Hilliard System

The Navier-Stokes-Cahn-Hilliard (NSCH) system governs the diffuse-interface dynamics of two incompressible and immiscible fluids. We consider a relaxation approximation of the NSCH system that is composed by a system of first-order hyperbolic balance laws and second-order elliptic operators. We prove first that the solutions of an initial boundary value problem for the approximation recover the limiting NSCH system for vanishing relaxation parameters. To cope with the singular limit we exploit the fact that the approximate solutions dissipate an almost quadratic energy, and employ the relative entropy-framework. In the second part of the work we provide numerical evidence for the analytical results, even in flow regimes not covered by the assumptions needed for the theoretical results. Using a novel marker-and-cell conservative finite-difference approach for both the approximation and the limit system, we are able to compute physically relevant interfacial flow problems including Ostwald ripening and high-velocity flow.

math.NA↗

arXiv2026

Thermodynamically consistent phase-field modeling and numerical simulation for reactive two-phase fluid-solid dynamics

We introduce a coupled Cahn-Hilliard Navier-Stokes model that governs the reactive two-phase dynamics of a system that consists of a fluid and a solid phase and prove its thermodynamic consistency. Moreover, we present an associated fully-discrete numerical method that relies on a continuous finite element approach and a semi-implicit time-stepping method. Our main theoretical result establishes that the fully discrete method satisfies an analog of the free energy dissipation inequality, provided that the phase-field variable remains within prescribed bounds. Numerical experiments confirm the theoretical findings and show the applicability of the method for realistic settings. In this context, we provide a preprocessing strategy that enables computing fluid flow in complex geometries given a sharp-interface formulation of the initial phase distribution. Moreover, we briefly introduce different solution strategies for the novel discretization based on the monolithic and partitioned solution paradigms and assess these in a comparative study.

math.NA↗

arXiv2025

Effective Equations for a Compressible Liquid-Vapor Flow Model with Highly Oscillating Initial Density

We derive and justify a new effective model for a compressible viscous liquid-vapor flow on a spray-like scale, i.e., for settings with a large number of phase boundaries. As a model on the detailed scale, we start from a parabolic relaxation of the Navier-Stokes-Korteweg system. We consider a sequence of initial data where the sequence of initial densities is assumed to be highly oscillating mimicking the high number of phase boundaries initially. Then, we consider a sequence of finite energy weak solutions corresponding to the sequence of initial data. Anticipating that the effective equations are found in the limit of infinitely many initial phase changes, we interpret the densities as Young measures and prove the convergence of the sequence of solutions to the effective model. The effective model consists of a deterministic part for the fluid's hydrodynamic quantities and a kinetic equation for the limit Young measure encoding the mixing dynamics. By characterizing the Young measure with the corresponding cumulative distribution function, we rewrite the kinetic equation for the Young measure into a kinetic equation for the cumulative distribution function such that the resulting equations are accessible by standard approximation methods.

math.AP↗

arXiv2025

Weak-Strong Uniqueness and Relaxation Limit for a Navier-Stokes-Korteweg Model

We consider a parabolic relaxation model for the compressible Navier-Stokes-Korteweg equations in the isothermal framework. This system depends on the relaxation parameters $\alpha,\beta>0$ and approximates formally solutions of the compressible Navier-Stokes-Korteweg equations in the relaxation limit $\alpha \to \infty$ and $\beta\to 0$. Introducing the class of finite energy weak solutions for the initial-boundary value problem corresponding to the relaxation model in spatial dimension three, we show that the weak-strong uniqueness principle holds. It asserts that a weak solution and a strong solution emanating from the same initial data coincide as long as the strong solution exists. Furthermore, we contribute a rigorous convergence result for the relaxation limit $\alpha \to \infty$ and $\beta\to 0$ and thus justify the relaxation model as an approximate model for the compressible Navier-Stokes-Korteweg equations from a mathematical point of view. Our results hold for general non-monotone pressure-density relations.

math.AP↗

arXiv2025

On hyperbolic approximations for a class of dispersive and diffusive-dispersive equations

We introduce novel approximate systems for dispersive and diffusive-dispersive equations with nonlinear fluxes. For purely dispersive equations, we construct a first-order, strictly hyperbolic approximation. Local well-posedness of smooth solutions is achieved by constructing a unique symmetrizer that applies to arbitrary smooth fluxes. Under stronger conditions on the fluxes, we provide a strictly convex entropy for the hyperbolic system that corresponds to the energy of the underlying dispersive equation. To approximate diffusive-dispersive equations, we rely on a viscoelastic damped system that is compatible with the found entropy for the hyperbolic approximation of the dispersive evolution. For the resulting hyperbolic-parabolic approximation, we provide a global well-posedness result. Using the relative entropy framework \cite{dafermos2005hyperbolic}, we prove that the solutions of the approximate systems converge to solutions of the original equations. The structure of the new approximate systems allows to apply standard numerical simulation methods from the field of hyperbolic balance laws. We confirm the convergence of our approximations even beyond the validity range of our theoretical findings on set of test cases covering different target equations. We show the applicability of the approach for strong nonlinear effects leading to oscillating or shock-layer-forming behavior.

math.AP↗

arXiv2025

Fractured Poroelastic Media in the Limit of Vanishing Aperture

We consider a poroelastic medium with a thin heterogeneity, also referred to as a fracture. Fluid flow and mechanical deformation inside both bulk and fracture are governed by the quasi-static Biot equations. The fracture's material parameters, such as hydraulic conductivity and elasticity, are assumed to scale with powers of the width-to-length ratio $\varepsilon$ of the fracture. Based on a priori estimates, we rigorously derive limit models as $\varepsilon \rightarrow 0$ and identify different limit regimes. We obtain five regimes for the hydraulic conductivity and two for the elasticity. While many cases yield discrete fracture models, others result in two-scale limit problems dominated by normal flow or deformation.

math.AP↗

arXiv2025

An Energy-Stable Discontinuous Galerkin Method for the Compressible Navier--Stokes--Allen--Cahn System

We consider a Navier--Stokes--Allen--Cahn (NSAC) system that governs the compressible motion of a viscous, immiscible two-phase fluid at constant temperature. Weak solutions of the NSAC system dissipate an appropriate energy functional. Based on an equivalent re-formulation of the NSAC system we propose a fully-discrete discontinuous Galerkin (dG) discretization that is mass-conservative, energy-stable, and provides higher-order accuracy in space and second-order accuracy in time. The approach relies on the approach in \cite{Giesselmann2015a} and a special splitting discretization of the derivatives of the free energy function within the Crank-Nicolson time-stepping. Numerical experiments confirm the analytical statements and show the applicability of the approach.

math.NA↗

arXiv2025

Existence and stability of the Riemann solutions for a non-symmetric Keyfitz--Kranzer type model

In this article, we develop a new hyperbolic model governing the first-order dynamics of a thin film flow under the influence of gravity and solute transport. The obtained system turns out to be a non-symmetric Keyfitz-Kranzer type system. We find an entire class of convex entropies in the regions where the system remains strictly hyperbolic. By including delta shocks, we prove the existence of unique solutions of the Riemann problem. We analyze their stability with respect to the perturbation of the initial data and to the gravity and surface tension parameters. Moreover, we discuss the large time behaviour of the solutions of the perturbed Riemann problem and prove that the initial Riemann states govern it. Thus, we confirm the structural stability of the Riemann solutions under the perturbation of initial data. Finally, we validate our analytical results with well-established numerical schemes for this new system of conservation laws.

math.AP↗

arXiv2025

Numerical approximations to statistical conservation laws for scalar hyperbolic equations

Motivated by the statistical description of turbulence, we study statistical conservation laws in the form of kinetic-type PDEs for joint probability density functions (PDFs) and cumulative distribution functions (CDFs) associated with solutions of scalar balance laws. Starting from viscous balance laws, the resulting PDF/CDF equations involve unclosed conditional averages arising in the viscous terms. We show that these terms exhibit a dissipative anomaly: they remain non-negligible in the vanishing viscosity limit and are essential to preserve the nonnegativity of evolving PDFs. To approximate these PDF/CDF equations in a unified framework, we propose a novel sampling-based estimator for the unclosed terms, constructed from numerical or exact realizations of the underlying balance-law solutions. In certain cases, a priori error bounds can be derived, demonstrating that the deviation between the true and approximate CDFs is controlled by the estimation error of the unclosed terms. Numerical experiments with analytically solvable test problems confirm that the sampling-based approximation converges satisfactorily with the number of samples.

math.NA↗

arXiv2025

Statistical conservation laws for scalar model problems: Hierarchical evolution equations

The probability density functions (PDFs) for the solution of the incompressible Navier-Stokes equation can be represented by a hierarchy of linear equations. This article develops new hierarchical evolution equations for PDFs of a scalar conservation law with random initial data as a model problem. Two frameworks are developed, including multi-point PDFs and single-point higher-order derivative PDFs. These hierarchies capture statistical correlations and guide closure strategies.

math.AP↗

arXiv2025

Entropy stable high-order discontinuous Galerkin spectral-element methods on curvilinear, hybrid meshes

Hyperbolic-parabolic partial differential equations are widely used for the modeling of complex, multiscale problems. High-order methods such as the discontinuous Galerkin (DG) scheme are attractive candidates for their numerical approximation. However, high-order methods are prone to instabilities in the presence of underresolved flow features. A popular counter measure to stabilize DG methods is the use of entropy-stable formulations based on summation-by-parts (SBP) operators. The present paper aims to construct a robust and efficient entropy-stable discontinuous Galerkin spectral element method (DGSEM) of arbitrary order on heterogeneous, curvilinear grids composed of triangular and quadrilateral elements or hexahedral, prismatic, tetrahedral and pyramid elements. To the author's knowledge, with the exception of hexahedral and quadrilateral elements, entropy-stable DGSE operators have been constructed exclusively for tetrahedral and triangular meshes. The extension of the DGSEM to more complex element shapes is achieved by means of a collapsed coordinate transformation. Legendre--Gauss quadrature nodes are employed as collocation points in conjunction with a generalized SBP operator and entropy-projected variables. The purely hyperbolic operator is extended to hyperbolic-parabolic problems by the use of a lifting procedure. To circumvent the penalizing time step restriction imposed by the collapsing, modal rather than nodal degrees of freedom are evolved in time, thereby relying on a memory-efficient weight-adjusted approximation to the inverse of the mass matrix. Essential properties of the proposed numerical scheme including free-stream preservation, polynomial and grid convergence as well as entropy conservation / stability are verified. Finally, with the flow around the common research model, the applicability of the presented method to real-world problems is demonstrated.

math.NA↗