Hendrik Ranocha

Hendrik Ranocha
University of Hamburg | UHH · Department of Mathematics

About

80
Publications
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856
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Introduction
Hendrik Ranocha is an Assistant Professor in Applied Mathematics at the University of Hamburg, Germany. His research is focused on the analysis and development of numerical methods for partial and ordinary differential equations. In particular, he is interested in the stability of these schemes and mimetic & structure-preserving techniques, allowing the transfer of results from the continuous level to the discrete one.
Additional affiliations
March 2021 - February 2022
University of Münster
Position
  • PostDoc Position
September 2019 - February 2021
King Abdullah University of Science and Technology
Position
  • PostDoc Position
April 2016 - September 2019
Technische Universität Braunschweig
Position
  • Research and Teaching Assistant

Publications

Publications (80)
Article
Full-text available
Entropy-conserving numerical fluxes are a cornerstone of modern high-order entropy-dissipative discretizations of conservation laws. In addition to entropy conservation, other structural properties mimicking the continuous level such as pressure equilibrium and kinetic energy preservation are important. This note proves that there are no numerical...
Preprint
Full-text available
High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable DG methods which incorporate an "entropy projec...
Article
Full-text available
We present Trixi.jl, a Julia package for adaptive high-order numerical simulations of hyperbolic partial differential equations. Utilizing Julia’s strengths, Trixi.jl is extensible, easy to use, and fast. We describe the main design choices that enable these features and compare Trixi.jl with a mature open source Fortran code that uses the same num...
Article
For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/ dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall ((2018) [1]). In this work, the correc...
Preprint
Full-text available
Entropy-conserving numerical fluxes are a cornerstone of modern high-order entropy-dissipative discretizations of conservation laws. In addition to entropy conservation, other structural properties mimicking the continuous level such as pressure equilibrium and kinetic energy preservation are important. This note proves that there are no numerical...
Preprint
Full-text available
Many modern discontinuous Galerkin (DG) methods for conservation laws make use of summation by parts operators and flux differencing to achieve kinetic energy preservation or entropy stability. While these techniques increase the robustness of DG methods significantly, they are also computationally more demanding than standard weak form nodal DG me...
Article
Full-text available
We study the numerical error in solitary wave solutions of nonlinear dispersive wave equations. A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy, but grows only linearly for conservative methods. W...
Preprint
Full-text available
We present BSeries.jl, a Julia package for the computation and manipulation of B-series, which are a versatile theoretical tool for understanding and designing discretizations of differential equations. We give a short introduction to the theory of B-series and associated concepts and provide examples of their use, including method composition and...
Article
Full-text available
We develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded...
Article
Full-text available
Many important differential equations model quantities whose value must remain positive or stay in some bounded interval. These bounds may not be preserved when the model is solved numerically. We propose to ensure positivity or other bounds by applying Runge–Kutta integration in which the method weights are adapted in order to enforce the bounds....
Article
Full-text available
SummationByPartsOperators.jl is a Julia library of summation-by-parts (SBP) operators, which are discrete derivative operators developed to get provably stable (semi-) discretizations, paying special attention to boundary conditions. Discretizations included in this framework are finite difference, Fourier pseudospectral, continuous Galerkin, and d...
Preprint
Patankar-type schemes are linearly implicit time integration methods designed to be unconditionally positivity-preserving by going outside of the class of general linear methods. Thus, classical stability concepts cannot be applied and there is no satisfying stability theory for these schemes. We develop a new approach to study stability properties...
Preprint
Full-text available
We present Trixi.jl, a Julia package for adaptive high-order numerical simulations of hyperbolic partial differential equations. Utilizing Julia's strengths, Trixi.jl is extensible, easy to use, and fast. We describe the main design choices that enable these features and compare Trixi.jl with a mature open source Fortran code that uses the same num...
Article
Full-text available
Recently, it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the simple density wave propagation example for the compressible Euler equations. The issue is related to missing local linear stability, i.e., the stability of the discretizati...
Article
Full-text available
We present a statistical survey of large-amplitude, asymmetric plasma and magnetic field enhancements detected outside the diamagnetic cavity at comet 67P/Churyumov–Gerasimenko from December 2014 to June 2016. Based on the concurrent observations of plasma and magnetic field enhancements, we interpret them to be magnetosonic waves. The aim is to pr...
Article
Full-text available
We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (...
Article
Full-text available
We study nonlinear hyperbolic conservation laws with non-convex flux in one space dimension and, for a broad class of numerical methods based on summation by parts operators, we compute numerically the kinetic functions associated with each scheme. As established by LeFloch and collaborators, kinetic functions (for continuous or discrete models) un...
Article
One of the challenges when simulating astrophysical flows with self-gravity is to compute the gravitational forces. In contrast to the hyperbolic hydrodynamic equations, the gravity field is described by an elliptic Poisson equation. We present a purely hyperbolic approach by reformulating the elliptic problem into a hyperbolic diffusion problem, w...
Preprint
Full-text available
We develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded...
Article
Full-text available
Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of A stable SBP time integration methods whic...
Preprint
Full-text available
We study the numerical error in solitary wave solutions of nonlinear dispersive wave equations. A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy, but grows only linearly for conservative methods. W...
Article
Full-text available
This paper proposes a new, conservative fully discrete scheme for the numerical solution of the regularized shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagati...
Preprint
Full-text available
We present a statistical survey of large amplitude, asymmetric plasma, and magnetic field enhancements at comet 67P/Churyumov-Gerasimenko from December 2014 to June 2016. The aim is to provide a general overview of these structures' properties over the mission duration. At comets, nonlinear wave evolution plays an integral part in the development o...
Article
Full-text available
Recently, an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial-value problems, using Runge–Kutta methods. We generalize this approach to multistep methods, including all general linear methods of order two or higher, and many other classes of schemes. We prove t...
Article
Full-text available
Ordinary differential equations (ODEs) are used to model a vast range of physical and other phenomena. They also arise in the discretization of partial differential equations. In most cases, solutions of differential equations must be approximated by numerical methods. The study of the properties of numerical methods for ODEs comprises an important...
Article
Full-text available
Ordinary and partial differential equations (ODEs and PDEs) are used to model many important phenomena. In most cases, solutions of these models must be approximated by numerical methods. Most of the relevant algorithms fall within a few classes of methods, with the properties of individual methods determined by their coefficients. The choice of ap...
Article
In computational fluid dynamics, the demand for increasingly multidisciplinary reliable simulations, for both analysis and design optimization purposes, requires transformational advances in individual components of future solvers. At the algorithmic level, hardware compatibility and efficiency are of paramount importance in determining viability a...
Preprint
Full-text available
Recently, it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the simple density wave propagation example for the compressible Euler equations. The issue is related to missing local linear stability, i.e. the stability of the discretizatio...
Preprint
Full-text available
The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagatio...
Article
Full-text available
Recently, relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation. Here, we generalize this approach to guarantee local entropy inequalities for finitely many convex functionals (entropies) and apply the resulting methods to the compressible Euler and Na...
Preprint
Full-text available
One of the challenges when simulating astrophysical flows with self-gravity is to compute the gravitational forces. In contrast to the hyperbolic hydrodynamic equations, the gravity field is described by an elliptic Poisson equation. We present a purely hyperbolic approach by reformulating the elliptic problem into a hyperbolic diffusion problem, w...
Preprint
Full-text available
We study nonlinear hyperbolic conservation laws with non-convex flux and, for a broad class of numerical methods based on summation by parts operators, we compute numerically the kinetic functions associated with each scheme. As established by LeFloch and collaborators, kinetic functions (for continuous or discrete models) uniquely characterize the...
Article
Full-text available
The recently-introduced relaxation approach for Runge–Kutta methods can be used to enforce conservation of energy in the integration of Hamiltonian systems. We study the behavior of implicit and explicit relaxation Runge–Kutta methods in this context. We find that, in addition to their useful conservation property, the relaxation methods yield othe...
Conference Paper
Full-text available
When considering hardware platforms, not just time-to-solution can be of importance but also the energy necessary to reach it. This is not only the case with battery powered mobile devices but also with HPC cluster systems due to financial and practical limits on power consumption and cooling. With a variety of hardware options available, the quest...
Preprint
Full-text available
We develop general tools to construct fully-discrete, conservative numerical methods and apply them to several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations conserve all linear invariants and one nonlinear invariant f...
Preprint
Full-text available
Many important differential equations model quantities whose value must remain positive or stay in some bounded interval. These bounds may not be preserved when the model is solved numerically. We propose to ensure positivity or other bounds by applying Runge-Kutta integration in which the method weights are adapted in order to enforce the bounds....
Article
Full-text available
Explicit Runge–Kutta methods are classical and widespread techniques in the numerical solution of ordinary differential equations (ODEs). Considering partial differential equations, spatial semidiscretizations can be used to obtain systems of ODEs that are solved subsequently, resulting in fully discrete schemes. However, certain stability investig...
Preprint
Full-text available
Recently, relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation. We generalize this approach to guarantee local entropy inequalities for finitely many convex functionals (entropies) and apply the resulting methods to the compressible Euler and Navier-S...
Preprint
Full-text available
When considering different hardware platforms, not just the time-to-solution can be of importance but also the energy necessary to reach it. This is not only the case with battery powered and mobile devices but also with high-performance parallel cluster systems due to financial and practical limits on power consumption and cooling. Recent developm...
Preprint
Full-text available
Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is necessarily involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of $A$ stable SBP time integratio...
Preprint
Full-text available
Recently, an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial-value problems, using Runge-Kutta methods. We generalize this approach to multistep methods, including all general linear methods of order two or higher, and many other classes of schemes. We prove t...
Article
Full-text available
In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations, which ar...
Preprint
Full-text available
The recently-introduced relaxation approach for Runge-Kutta methods can be used to enforce conservation of energy in the integration of Hamiltonian systems. We study the behavior of implicit and explicit relaxation Runge-Kutta methods in this context. We find that, in addition to their useful conservation property, the relaxation methods yield othe...
Article
Full-text available
The framework of inner product norm preserving relaxation Runge-Kutta methods [D. I. Ketcheson, SIAM J. Numer. Anal., 57 (2019), pp. 2850-2870] is extended to general convex quantities. Conservation, dissipation, or other solution properties with respect to any convex functional are enforced by the addition of a relaxation parameter that multiplies...
Chapter
Full-text available
High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions used in both the finite difference and the discontinuous Galerkin spectral element framework. I...
Article
Full-text available
Many important initial value problems have the property that energy is nonincreasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for conditional energy stability of explicit Runge--Kutta methods for nonlinear or nonautonomous problems. We provide bo...
Preprint
Full-text available
In computational fluid dynamics, the demand for increasingly multidisciplinary reliable simulations, for both analysis and design optimization purposes, requires transformational advances in individual components of future solvers. At the algorithmic level, hardware compatibility and efficiency are of paramount importance in determining viability a...
Preprint
Full-text available
In computational fluid dynamics, the demand for increasingly multidisciplinary reliable simulations, for both analysis and design optimization purposes, requires transformational advances in individual components of future solvers. At the algorithmic level, hardware compatibility and efficiency are of paramount importance in determining viability a...
Preprint
Full-text available
Many important initial value problems have the property that energy is non-increasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for conditional energy stability of explicit Runge-Kutta methods for nonlinear or non-autonomous problems. We provide b...
Preprint
Full-text available
In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations, which ar...
Preprint
For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative semidiscretisations by adding suitable correction terms has been proposed recently by Abgrall (J. Comp. Phys. 372: pp. 640-666, 2018). H...
Article
Full-text available
For practical applications, the long time behaviour of the error of numerical solutions to time-dependent partial differential equations is very important. Here, we investigate this topic in the context of hyperbolic conservation laws and flux reconstruction schemes, focusing on the schemes in the discontinuous Galerkin spectral element framework....
Preprint
Full-text available
The framework of inner product norm preserving relaxation Runge-Kutta methods (David I. Ketcheson, Relaxation Runge-Kutta Methods: Conservation and Stability for Inner-Product Norms, 2019) is extended to general convex quantities. Conservation, dissipation, or other solution properties with respect to any convex functional are enforced by the addit...
Article
Full-text available
Some properties of numerical time integration methods using summation by parts (SBP)operators and simultaneous approximation terms are studied. These schemes can be interpreted as implicit Runge-Kutta methods with desirable stability properties such as A-, B-, L-, and algebraic stability [1–4]. Here, insights into the necessity of certain assumptio...
Preprint
Full-text available
Some properties of numerical time integration methods using summation by parts operators and simultaneous approximation terms are studied. These schemes can be interpreted as implicit Runge-Kutta methods with desirable stability properties such as $A$-, $B$-, $L$-, and algebraic stability. Here, insights into the necessity of certain assumptions, r...
Preprint
Full-text available
Explicit Runge-Kutta methods are classical and widespread techniques in the numerical solution of ordinary differential equations (ODEs). Considering partial differential equations, spatial semidiscretisations can be used to obtain systems of ODEs that are solved subsequently, resulting in fully discrete schemes. However, certain stability investig...
Article
Full-text available
For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear systems in several space dimensions is an open problem, mimetic properties such as summation-by-parts as discret...
Preprint
The nonlinear magnetic induction equation with Hall effect can be used to model magnetic fields, e.g. in astrophysical plasma environments. In order to give reliable results, numerical simulations should be carried out using effective and efficient schemes. Thus, high-order stable schemes are investigated here. Following the approach provided recen...
Article
Full-text available
Entropy stability is a well-known design principle for numerical methods in gas dynamics. Entropy conservative numerical fluxes can be used as ingredients in two kinds of schemes: Firstly, as building blocks in the subcell flux differencing form of Fisher and Carpenter (2013) and secondly (enhanced by dissipation) as numerical surface fluxes in fin...
Chapter
The correction procedure via reconstruction (CPR, also known as flux reconstruction), is a high-order numerical scheme for conservation laws introduced by Huynh (2007), unifying some discontinuous Galerkin, spectral difference and spectral volume methods. A general framework of summation-by-parts (SBP) operators with simultaneous approximation term...
Chapter
We focus on spectral viscosity in the framework of correction procedure via reconstruction (CPR, also known as flux reconstruction) using summation-by-parts (SBP) operators. In Ranocha et al. (J Comput Phys 342:13–28, 2017), [10], Ranocha et al. (J Comput Phys 311:299–328, 2016), [9], the authors used SBP operators in the CPR framework and were abl...
Preprint
For practical applications, the long time behaviour of the error of numerical solutions to time-dependent partial differential equations is very important. Here, we investigate this topic in the context of hyperbolic conservation laws and flux reconstruction schemes. For linear problems with constant coefficients, it is well-known in the literature...
Article
The flux reconstruction is a framework of high order semidiscretisations used for the numerical solution of hyperbolic conservation laws. Using a reformulation of these schemes relying on summation-by-parts (SBP) operators and simultaneous approximation terms, artificial dissipation/spectral viscosity operators and connections to modal filtering ar...
Preprint
Full-text available
For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear systems in several space dimensions is an open problem, mimetic properties such as summation-by-parts as discret...
Conference Paper
Full-text available
OpenCL offers many advantages in computational physics in comparison to traditional MPI/OpenMP parallelization. We present an MPI/OpenCL based plasma simulation code, as an example of how computational physics can benefit from OpenCL. The code utilizes a hybrid modeling approach which combines elements from both fluid and particle-in-cell (PIC) met...
Thesis
It has been known for a long time that high order methods can be very efficient for the numerical solution of hyperbolic balance laws if their stability is ensured. A suitable way to investigate stability of linear partial differential equations is given by the energy method, relying on integration by parts. In order to transfer these methods to th...
Article
Explicit Runge–Kutta methods are standard tools in the numerical solution of ordinary differential equations (ODEs). Applying the method of lines to partial differential equations, spatial semidiscretisations result in large systems of ODEs that are solved subsequently. However, stability investigations of high-order methods for transport equations...
Article
Full-text available
High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finit...
Article
Full-text available
Entropy stable semidiscretisations of the shallow water equations are developed, based on summation-by-parts (SBP) operators and using split forms of the equations. The resulting two-parameter family of entropy conservative schemes for general SBP bases, especially using Gau{\ss} nodes, is adapted to varying bottom topography in a well-balanced way...
Article
A generalised analytical notion of summation-by-parts (SBP) methods is proposed, extending the concept of SBP operators in the correction procedure via reconstruction (CPR), a framework of high-order methods for conservation laws. For the first time, SBP operators with dense norms and not including boundary points are used to get an entropy stable...
Article
Full-text available
In this paper, we consider Burgers' equation with uncertain boundary and initial conditions. The polynomial chaos (PC) approach yields a hyperbolic systems of deterministic equations, which can be solved by several numerical methods. Here, we apply the correction procedure via reconstruction (CPR) using summation-by-parts operators. We focus especi...
Article
Full-text available
Splitting the discretisation of a partial differential equation by the method of lines in a semidiscretisation in space and a following integration in time, the coupling of these schemes has to be considered for fully discrete stability. This article focuses at first on the extension of adaptive techniques developed by the authors for semidiscretis...
Article
Full-text available
A recently introduced framework of semidiscretisations for hyperbolic conservation laws known as correction procedure via reconstruction (CPR, also known as flux reconstruction) is considered in the extended setting of summation-by-parts (SBP) operators using simultaneous approximation terms (SATs). This reformulation can yield stable semidiscretis...
Article
Full-text available
The correction procedure via reconstruction (CPR, also known as flux reconstruction) is a framework of high order semidiscretisations used for the numerical solution of hyperbolic conservation laws. Using a reformulation of these schemes relying on summation-by-parts (SBP) operators and simultaneous approximation terms (SATs), artificial dissipatio...
Article
Full-text available
A generalised analytical notion of summation-by-parts (SBP) methods is proposed, extending the concept of SBP operators in the correction procedure via reconstruction (CPR), a framework of high-order methods for conservation laws. For the first time, SBP operators with dense norms and not including boundary points are used to get an entropy stable...
Article
The correction procedure via reconstruction (CPR, formerly known as flux reconstruction) is a framework of high order methods for conservation laws, unifying some discontinuous Galerkin, spectral difference and spectral volume methods. Linearly stable schemes were presented by Vincent et al. (2011, 2015), but proofs of non-linearly (entropy) stabil...
Article
In the advent of the Rosetta arrival at the comet 67P/Churyumov-Gerasimenko, we present a global 3D hybrid simulation model of the cometary plasma interaction which resolves the innermost coma sufficiently. As Rosetta will only provide local information, global simulations are required to put these local observations into a wider global perspective...

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