arXiv subjects

Hendrik Ranocha

Publications and source records attributed to Hendrik Ranocha.

At least 37 records · Page 2Linked to original sources

arXiv2026

Structure-preserving approximations of the Serre-Green-Naghdi equations in standard and hyperbolic form

We develop structure-preserving numerical methods for the Serre-Green-Naghdi equations, a model for weakly dispersive free-surface waves. We consider both the classical form, requiring the inversion of a non-linear elliptic operator, and a hyperbolic approximation of the equations, allowing fully explicit time stepping. Systems for both flat and variable topography are studied. Our novel numerical methods conserve both the total water mass and the total energy. In addition, the methods for the original Serre-Green-Naghdi equations conserve the total momentum for flat bathymetry. For variable topography, all the methods proposed are well-balanced for the lake-at-rest state. We provide a theoretical setting allowing us to construct schemes of any kind (finite difference, finite element, discontinuous Galerkin, spectral, etc.) as long as summation-by-parts operators are available in the chosen setting. Energy-stable variants are proposed by adding a consistent high-order artificial viscosity term. The proposed methods are validated through a large set of benchmarks to verify all the theoretical properties. Whenever possible, comparisons with exact, reference numerical, or experimental data are carried out. The impressive advantage of structure preservation, and in particular energy preservation, to resolve accurately dispersive wave propagation on very coarse meshes is demonstrated by several of the tests.

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arXiv2026

Compact Runge-Kutta flux reconstruction methods with entropy and/or kinetic energy preserving fluxes

Compact Runge-Kutta (cRK) methods are a class of high order methods for solving hyperbolic conservation laws characterized by their compact stencil including only immediate neighboring finite elements. A Compact Runge-Kutta flux reconstruction (cRKFR) method for solver hyperbolic conservation laws was introduced in [Babbar, A., Chen, Q., Journal of Scientific Computing, 2025] which uses a time average flux formulation to perform evolution using a single numerical flux computation at each step, making it a single stage method. Entropy or kinetic energy preserving numerical fluxes are often used for construction of high order entropy stable or kinetic energy preserving methods for hyperbolic conservation laws, and are known to enhance the robustness of numerical methods for under-resolved simulations. In this work, we show how these fluxes can be incorporated into the cRKFR framework for general hyperbolic equations that consist of fluxes and non-conservative products. We test the effectiveness of this new class of methods through numerical experiments for the compressible Euler equations, magnetohydronamics (MHD) equations and multi-ion MHD equations. It is observed that the application of entropy or kinetic energy preserving fluxes enhances the robustness of the cRKFR methods.

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arXiv2026

A Positivity-Preserving Relaxation Algorithm

We combine Patankar-type methods with suitable relaxation procedures that are capable of ensuring correct dissipation or conservation of functionals such as entropy or energy while producing unconditionally positive and conservative approximations. To that end, we adapt the relaxation algorithm to enforce positivity by using either ideas from the dense output framework when a linear invariant must be preserved, or simply a geometric mean if the only constraint is positivity preservation. The latter merely requires the solution of a scalar nonlinear equation while former results in a coupled linear-nonlinear system of equations. We present sufficient conditions for the solvability of the respective equations. Several applications in the context of ordinary and partial differential equations are presented, and the theoretical findings are validated numerically.

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arXiv2026

On Affordable High-Order Entropy-Conservative/Stable and Well-Balanced Methods for Nonconservative Hyperbolic Systems

Many entropy-conservative and entropy-stable (summarized as entropy-preserving) methods for hyperbolic conservation laws rely on Tadmor's theory for two-point entropy-preserving numerical fluxes and its higher-order extension via flux differencing using summation-by-parts (SBP) operators, e.g., in discontinuous Galerkin spectral element methods (DGSEMs). The underlying two-point formulations have been extended to nonconservative systems using fluctuations by Castro et al. (2013, doi:10.1137/110845379) with follow-up generalizations to SBP methods. We propose specific forms of entropy-preserving fluctuations for nonconservative hyperbolic systems that are simple to interpret and allow an algorithmic construction of entropy-preserving methods. We analyze necessary and sufficient conditions, and obtain a full characterization of entropy-preserving three-point methods within the finite volume framework. This formulation is extended to SBP methods in multiple space dimensions on Cartesian and curvilinear meshes. Additional properties such as well-balancedness extend naturally from the underlying finite volume method to the SBP framework. We use the algorithmic construction enabled by the chosen formulation to derive several new entropy-preserving schemes for nonconservative hyperbolic systems, e.g., the compressible Euler equations of an ideal gas using the internal energy equation and a dispersive shallow-water model. Numerical experiments show the robustness and accuracy of the proposed schemes.

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arXiv2026

Volume Term Adaptivity for Discontinuous Galerkin Schemes

We introduce the concept of volume term adaptivity for high-order discontinuous Galerkin (DG) schemes solving time-dependent partial differential equations. Termed v-adaptivity, we present a novel general approach that exchanges the discretization of the volume contribution of the DG scheme at every Runge-Kutta stage based on suitable indicators. Depending on whether robustness or efficiency is the main concern, different adaptation strategies can be chosen. Precisely, the weak form volume term discretization is used instead of the entropy-conserving flux-differencing volume integral whenever the former produces more entropy than the latter, resulting in an entropy-stable scheme. Conversely, if increasing the efficiency is the main objective, the weak form volume integral may be employed as long as it does not increase entropy beyond a certain threshold or cause instabilities. Thus, depending on the choice of the indicator, the v-adaptive DG scheme improves robustness, efficiency and approximation quality compared to schemes with a uniform volume term discretization. We thoroughly verify the accuracy, linear stability, and entropy-admissibility of the v-adaptive DG scheme before applying it to various compressible flow problems in two and three dimensions.

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arXiv2026

Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes

Jin-Xin relaxation is a method for approximating non-linear hyperbolic conservation laws by a linear system of hyperbolic equations with an $\varepsilon$ dependent stiff source term. The system formally relaxes to the original conservation law as $\varepsilon \to 0$. An asymptotic analysis of the Jin-Xin relaxation system shows that it can be seen as a convection-diffusion equation with a diffusion coefficient that depends on the relaxation parameter $\varepsilon$. This work makes use of this property to use the Jin-Xin relaxation system as a shock-capturing method for high-order discontinuous Galerkin (DG) or flux reconstruction (FR) schemes. The idea is to use a smoothness indicator to choose the $\varepsilon$ value in each cell, so that we can use larger $\varepsilon$ values in non-smooth regions to add extra numerical dissipation. We show how this can be done by using a single stage method by using the compact Runge-Kutta FR method that handles the stiff source term by using IMplicit-EXplicit Runge-Kutta (IMEX-RK) schemes. Numerical results involving Burgers' equation and the compressible Euler equations are shown to demonstrate the effectiveness of the proposed method.

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arXiv2026

Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions

We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the limiting problem exists. We only require weak (entropy) solutions of the hyperbolic approximations. Thereby, we provide a solid foundation for these approximations, which have been used in the literature without rigorous convergence analysis. We also present numerical results that support our theoretical findings.

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arXiv2026

An Energy-Preserving Domain of Dependence Stabilization for the Linear Wave Equation on Cut-Cell Meshes

We present an energy-preserving (either energy-conservative or energy-dissipative) domain of dependence stabilization method for the linear wave equation on cut-cell meshes. Our scheme is based on a standard discontinuous Galerkin discretization in space and an explicit (strong stability preserving) Runge Kutta method in time. Tailored stabilization terms allow for selecting the time step length based on the size of the background cells rather than the small cut cells by propagating information across small cut cells. The stabilization terms preserve the energy stability or energy conservation property of the underlying discontinuous Galerkin space discretization. Numerical results display the high accuracy and stability properties of our scheme.

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arXiv2026

Domain-of-dependence-stabilized cut-cell discretizations of linear kinetic models with summation-by-parts properties

We employ the summation-by-parts (SBP) framework to extend the recent domain-of-dependence (DoD) stabilization for cut cells to linear kinetic models in diffusion scaling. Numerical methods for these models are challenged by increased stiffness for small scaling parameters and the necessity of asymptotics preservation regarding a parabolic limit equation. As a prototype model, we consider the telegraph equation in one spatial dimension subject to periodic boundary conditions with an asymptotic limit given by the linear heat equation. We provide a general semidiscrete stability result for this model when spatially discretized by arbitrary periodic (upwind) SBP operators and formally prove that the fully discrete scheme is asymptotic preserving. Moreover, we prove that DoD with central numerical fluxes leads to periodic SBP operators. Furthermore, we show that adapting the upwind DoD scheme yields periodic upwind SBP operators. Consequently, the DoD stabilization possesses the desired properties considered in the first part of this work and thus leads to a stable and asymptotic preserving scheme for the telegraph equation. We back our theoretical results with numerical simulations and demonstrate the applicability of this cut-cell stabilization for implicit time integration in the heat equation limit.

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arXiv2025

Using Machine Learning to Design Time Step Size Controllers for Stable Time Integrators

We present a new method for developing time step controllers based on a technique from the field of machine learning. This method is applicable to stable time integrators that have an embedded scheme, i.e., that have local error estimation similar to Runge-Kutta pairs. To design good time step size controllers using these error estimates, we propose to use Bayesian optimization. In particular, we design a novel objective function that captures important properties such as tolerance convergence and computational stability. We apply our new approach to several modified Patankar--Runge--Kutta (MPRK) schemes and a Rosenbrock-type scheme, equipping them with controllers based on digital signal processing which extend classical PI and PID controllers. We demonstrate that the optimization process yields controllers that are at least as good as the best controllers chosen from a wide range of suggestions available for classical explicit and implicit time integration methods by providing work-precision diagrams on a variety of ordinary and partial differential equations.

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arXiv2025

Asymptotic-preserving and energy-conserving methods for a hyperbolic approximation of the BBM equation

We study the hyperbolic approximation of the Benjamin-Bona-Mahony (BBM) equation proposed recently by Gavrilyuk and Shyue (2022). We develop asymptotic-preserving numerical methods using implicit-explicit (additive) Runge-Kutta methods that are implicit in the stiff linear part. The new discretization of the hyperbolization conserves important invariants converging to invariants of the BBM equation. We use the entropy relaxation approach to make the fully discrete schemes energy-preserving. Numerical experiments demonstrate the effectiveness of these discretizations.

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arXiv2025

Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations

We use the general framework of summation-by-parts operators to construct conservative, energy-stable, and well-balanced semidiscretizations of two different nonlinear systems of dispersive shallow water equations with varying bathymetry: (i) a variant of the coupled Benjamin-Bona-Mahony (BBM) equations and (ii) a recently proposed model by Svärd and Kalisch (2025) with enhanced dispersive behavior. Both models share the property of being conservative in terms of a nonlinear invariant, often interpreted as energy. This property is preserved exactly in our novel semidiscretizations. To obtain fully-discrete energy-stable schemes, we employ the relaxation method. Our novel methods generalize energy-conserving methods for the BBM-BBM system to variable bathymetries. Compared to the low-order, energy-dissipative finite volume method proposed by Svärd and Kalisch, our schemes are arbitrary high-order accurate, energy-conservative or -stable, can deal with periodic and reflecting boundary conditions, and can be any method within the framework of summation-by-parts operators including finite difference and finite element schemes. We present improved numerical properties of our methods in some test cases.

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arXiv2025

Traveling-wave solutions and structure-preserving numerical methods for a hyperbolic approximation of the Korteweg-de Vries equation

We study the recently-proposed hyperbolic approximation of the Korteweg-de Vries equation (KdV). We show that this approximation, which we call KdVH, possesses a rich variety of solutions, including solitary wave solutions that approximate KdV solitons, as well as other solitary and periodic solutions that are related to higher-order water wave models, and may include singularities. We analyze a class of implicit-explicit Runge-Kutta time discretizations for KdVH that are asymptotic preserving, energy conserving, and can be applied to other hyperbolized systems. We also develop structure-preserving spatial discretizations based on summation-by-parts operators in space including finite difference, discontinuous Galerkin, and Fourier methods. We use the entropy relaxation approach to make the fully discrete schemes energy-preserving. Numerical experiments demonstrate the effectiveness of these discretizations.

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arXiv2025

Real-time rail vehicle localisation using spatially resolved magnetic field measurements

This work presents two complementary real-time rail vehicle localization methods based on magnetic field measurements and a pre-recorded magnetic map. The first uses a particle filter reweighted via magnetic similarity, employing a heavy-tailed non-Gaussian kernel for enhanced stability. The second is a stateless sequence alignment technique that transforms real-time magnetic signals into the spatial domain and matches them to the map using a similarity measure. Experiments with operational train data show that the particle filter achieves track-selective, sub-5-meter accuracy over 21.6 km, though its performance degrades at low speeds and during cold starts. Accuracy tests were constrained by the GNSS-based reference system. In contrast, the alignment-based method excels in cold-start scenarios, localizing within 30 m in 92 % of tests (100 % using top-3 matches). A hybrid approach combines both methods$\unicode{x2014}$alignment-based initialization followed by particle filter tracking. Runtime analysis confirms real-time capability on consumer-grade hardware. The system delivers accurate, robust localization suitable for safety-critical rail applications.

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arXiv2025

Paired Explicit Relaxation Runge-Kutta Methods: Entropy-Conservative and Entropy-Stable High-Order Optimized Multirate Time Integration

We present novel entropy-conservative and entropy-stable multirate Runge-Kutta methods based on Paired Explicit Runge-Kutta (P-ERK) schemes with relaxation for conservation laws and related systems of partial differential equations. Optimized schemes up to fourth-order of accuracy are derived and validated in terms of order of consistency, conservation of linear invariants, and entropy conservation/stability. We demonstrate the effectiveness of these P-ERRK methods when combined with a high-order, entropy-conservative/stable discontinuous Galerkin spectral element method on unstructured meshes. The Paired Explicit Relaxation Runge-Kutta methods(P-ERRK) are readily implemented for partitioned semidiscretizations arising from problems with equation-based scale separation such as non-uniform meshes. We highlight that the relaxation approach acts as a time-limiting technique which improves the nonlinear stability and thus robustness of the multirate schemes. The P-ERRK methods are applied to a range of problems, ranging from compressible Euler over compressible Navier-Stokes to the visco-resistive magnetohydrodynamics equations in two and three spatial dimensions. For each test case, we compare computational load and runtime to standalone relaxed Runge-Kutta methods which are outperformed by factors up to four. All results can be reproduced using a publicly available repository.

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arXiv2025

Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods

High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often incorporate stabilization techniques that must be applied judiciously -- sufficient to ensure simulation stability but restrained enough to prevent excessive dissipation and loss of resolution. Recent studies have demonstrated that combining upwind summation-by-parts (USBP) operators with flux vector splitting can increase the robustness of finite difference (FD) schemes without introducing excessive artificial dissipation. This work investigates whether the same approach can be applied to nodal discontinuous Galerkin (DG) methods. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our discussion encompasses a broad class of USBP operators, not limited to equidistant grid points, and enables the development of novel USBP operators on Legendre--Gauss--Lobatto (LGL) points that are well-suited for nodal DG methods. We then examine the robustness properties of the resulting DG-USBP methods for challenging examples of the compressible Euler equations, such as the Kelvin--Helmholtz instability. Similar to high-order FD-USBP schemes, we find that combining flux vector splitting techniques with DG-USBP operators does not lead to excessive artificial dissipation. Furthermore, we find that combining lower-order DG-USBP operators on three LGL points with flux vector splitting indeed increases the robustness of nodal DG methods. However, we also observe that higher-order USBP operators offer less improvement in robustness for DG methods compared to FD schemes. We provide evidence that this can be attributed to USBP methods adding dissipation only to unresolved modes...

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arXiv2025

Pseudo-Energy-Preserving Explicit Runge-Kutta Methods

Using a recent characterization of energy-preserving B-series, we derive the explicit conditions on the coefficients of a Runge-Kutta method that ensure energy preservation (for Hamiltonian systems) up to a given order in the step size, which we refer to as the pseudo-energy-preserving (PEP) order. We study explicit Runge-Kutta methods with PEP order higher than their classical order. We provide examples of such methods up to PEP order six, and test them on Hamiltonian ODE and PDE systems. We find that these methods behave similarly to exactly energy-conservative methods over moderate time intervals and exhibit significantly smaller errors, relative to other Runge-Kutta methods of the same order, for moderately long-time simulations.

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arXiv2024

Structure-Preserving Numerical Methods for Fokker-Planck Equations

A common way to numerically solve Fokker-Planck equations is the Chang-Cooper method in space combined with one of the Euler methods in time. However, the explicit Euler method is only conditionally positive, leading to severe restrictions on the time step to ensure positivity. On the other hand, the implicit Euler method is robust but nonlinearly implicit. Instead, we propose to combine the Chang-Cooper method with unconditionally positive Patankar-type time integration methods, since they are unconditionally positive, robust for stiff problems, only linearly implicit, and also higher-order accurate. We describe the combined approach, analyse it, and present a relevant numerical example demonstrating advantages compared to schemes proposed in the literature.

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