Jekyll2020-09-22T10:59:17+00:00https://ranocha.de/feed.xmlDr. Hendrik RanochaDr. Hendrik Ranocha is a Postdoctoral Fellow in the [group of David I. Ketcheson](http://numerics.kaust.edu.sa/) at [KAUST (King Abdullah University of Science and Technology, Saudi Arabia)](https://www.kaust.edu.sa/en). Before, he has been a member of the group of [Thomas Sonar](https://www.tu-braunschweig.de/icm/pde/personal/sonar/) in Braunschweig, 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.Dr. Hendrik Ranochamail@ranocha.deNew Paper ‘A conservative fully-discrete numerical method for the regularised shallow water wave equations’ on arXiv2020-09-22T00:00:00+00:002020-09-22T00:00:00+00:00https://ranocha.de/blog/BBM_BBM_FEM_arXiv<p>Dimitrios Mitsotakis, David I Ketcheson, Endre Süli, and I have published our new paper
<a href="https://arxiv.org/abs/2009.09641">A conservative fully-discrete numerical method for the regularised shallow water wave equations</a>
on arXiv.</p>
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<p>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 propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge-Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to the standard, non-conservative Galerkin methods.</p>
</blockquote>Dr. Hendrik Ranochamail@ranocha.deDimitrios Mitsotakis, David I Ketcheson, Endre Süli, and I have published our new paper A conservative fully-discrete numerical method for the regularised shallow water wave equations on arXiv.New Paper ‘A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics’ on arXiv2020-08-25T00:00:00+00:002020-08-25T00:00:00+00:00https://ranocha.de/blog/euler_gravity_arXiv<p>Michael Schlottke-Lakemper, Andrew R. Winters, Gregor J. Gassner, and I have published our new paper
<a href="https://arxiv.org/abs/2008.10593">A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics</a>
on arXiv.</p>
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<p>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, which is solved in pseudotime using the same explicit high-order discontinuous Galerkin method we use for the flow solution. The flow and the gravity solvers operate on a joint hierarchical Cartesian mesh and are two-way coupled via the source terms. A key benefit of our approach is that it allows the reuse of existing explicit hyperbolic solvers without modifications, while retaining their advanced features such as non-conforming and solution-adaptive grids. By updating the gravitational field in each Runge-Kutta stage of the hydrodynamics solver, high-order convergence is achieved even in coupled multi-physics simulations. After verifying the expected order of convergence for single-physics and multi-physics setups, we validate our approach by a simulation of the Jeans gravitational instability. Furthermore, we demonstrate the full capabilities of our numerical framework by computing a self-gravitating Sedov blast with shock capturing in the flow solver and adaptive mesh refinement for the entire coupled system.</p>
</blockquote>Dr. Hendrik Ranochamail@ranocha.deMichael Schlottke-Lakemper, Andrew R. Winters, Gregor J. Gassner, and I have published our new paper A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics on arXiv.Conference Proceedings ‘Entropy Conserving and Kinetic Energy Preserving Numerical Methods for the Euler Equations Using Summation-by-Parts Operators’ of ICOSAHOM 2018 published2020-08-16T00:00:00+00:002020-08-16T00:00:00+00:00https://ranocha.de/blog/ICOSAHOM_2018<p>My conference paper
<a href="https://doi.org/10.1007/978-3-030-39647-3_42">Entropy Conserving and Kinetic Energy Preserving Numerical Methods for the Euler Equations Using Summation-by-Parts Operators</a>
has been published.</p>
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<p>Summation-by-parts operators can be used in the context of finite difference and discontinuous Galerkin methods to create discretisations mimicking properties given at the continuous level such as entropy conservation. Recently, there has been some interest in schemes for the Euler equations that additionally preserve the kinetic energy. However, some these methods resulted in undesired and unexpected changes of the kinetic energy in numerical experiments of Gassner et al. (J Comput Phys 327:39–66, 2016). Here, analytical insights into kinetic energy preservation are given and new entropy conservative and kinetic energy preserving numerical fluxes are proposed.</p>
</blockquote>Dr. Hendrik Ranochamail@ranocha.deMy conference paper Entropy Conserving and Kinetic Energy Preserving Numerical Methods for the Euler Equations Using Summation-by-Parts Operators has been published.New Paper ‘Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts’ on arXiv2020-07-17T00:00:00+00:002020-07-17T00:00:00+00:00https://ranocha.de/blog/kinetic_functions_arXiv<p>Philippe G. LeFloch and I have published our new paper
<a href="https://arxiv.org/abs/2007.08780">Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts</a>
on arXiv.</p>
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<p>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 macro-scale dynamics of small-scale dependent, undercompressive, nonclassical shock waves. We show here that various entropy-dissipative numerical schemes can yield nonclassical solutions containing classical shocks, including Fourier methods with (super-) spectral viscosity, finite difference schemes with artificial dissipation, discontinuous Galerkin schemes with or without modal filtering, and TeCNO schemes. We demonstrate numerically that entropy stability does not imply uniqueness of the limiting numerical solutions, and we compute the associated kinetic functions in order to distinguish between these schemes. In addition, we design entropy-dissipative schemes for the Keyfitz-Kranzer system whose solutions are measures with delta shocks. This system illustrates the fact that entropy stability does not imply boundedness under grid refinement.</p>
</blockquote>Dr. Hendrik Ranochamail@ranocha.dePhilippe G. LeFloch and I have published our new paper Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts on arXiv.New Paper ‘Fully-Discrete Explicit Locally Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations’ published in Computers and Mathematics with Applications2020-07-12T00:00:00+00:002020-07-12T00:00:00+00:00https://ranocha.de/blog/local_relaxation<p>The paper
<a href="https://dx.doi.org/10.1016/j.camwa.2020.06.016">Fully-Discrete Explicit Locally Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations</a>
of Lisandro Dalcin, Matteo Parsani, and me has been published in
Computers and Mathematics with Applications.
As usual, you can <a href="https://arxiv.org/abs/2003.08831">find the preprint on arXiv</a>.</p>
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<p>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-Stokes equations. Based on the unstructured <em>hp</em>-adaptive SSDC framework of entropy conservative or dissipative semidiscretizations using summation-by-parts and simultaneous-approximation-term operators, we develop the first discretizations for compressible computational fluid dynamics that are primary conservative, locally entropy stable in the fully discrete sense under a usual CFL condition, explicit except for the parallelizable solution of a single scalar equation per element, and arbitrarily high-order accurate in space and time. We demonstrate the accuracy and the robustness of the fully-discrete explicit locally entropy-stable solver for a set of test cases of increasing complexity.</p>
</blockquote>Dr. Hendrik Ranochamail@ranocha.deThe paper Fully-Discrete Explicit Locally Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations of Lisandro Dalcin, Matteo Parsani, and me has been published in Computers and Mathematics with Applications. As usual, you can find the preprint on arXiv.New Paper ‘Relaxation Runge-Kutta Methods for Hamiltonian Problems’ published in Journal of Scientific Computing2020-07-10T00:00:00+00:002020-07-10T00:00:00+00:00https://ranocha.de/blog/Hamiltonian_RRK<p>The paper
<a href="https://dx.doi.org/10.1007/s10915-020-01277-y">Relaxation Runge-Kutta Methods for Hamiltonian Problems</a>
of David I. Ketcheson and me has been published in
Journal of Scientific Computing.
As usual, you can <a href="https://arxiv.org/abs/2001.04826">find the preprint on arXiv</a>.</p>
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<p>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 other improvements. Experiments show that their solutions bear stronger qualitative similarity to the true solution and that the error grows more slowly in time. We also prove that these methods are superconvergent for a certain class of Hamiltonian systems.</p>
</blockquote>Dr. Hendrik Ranochamail@ranocha.deThe paper Relaxation Runge-Kutta Methods for Hamiltonian Problems of David I. Ketcheson and me has been published in Journal of Scientific Computing. As usual, you can find the preprint on arXiv.Conference Proceedings ‘Towards Green Computing: A Survey of Performance and Energy Efficiency of Different Platforms using OpenCL’ published2020-06-29T00:00:00+00:002020-06-29T00:00:00+00:00https://ranocha.de/blog/Energy_efficiency_ACM<p>Philip Heinisch, Katharina Ostaszewski, and I have published our contribution
<a href="https://dx.doi.org/10.1145/3388333.3403035">Towards Green Computing: A Survey of Performance and Energy Efficiency of Different Platforms using OpenCL</a>
in the proceedings of the International Workshop on OpenCL IWOCL ‘20.
As usual, you can <a href="https://arxiv.org/abs/2003.03794">find the preprint and extended information on arXiv</a>.</p>
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<p>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 developments in hard- and software have given programmers the ability to run the same code on a range of different devices giving rise to the concept of heterogeneous computing. Many of these devices are optimized for certain types of applications. To showcase the differences and give a basic outlook on the applicability of different architectures for specific problems, the cross-platform OpenCL framework was used to compare both time- and energy-to-solution. A large set of devices ranging from ARM processors to server CPUs and consumer and enterprise level GPUs has been used with different benchmarking testcases taken from applied research applications. While the results show the overall advantages of GPUs in terms of both runtime and energy efficiency compared to CPUs, ARM devices show potential for certain applications in massively parallel systems. This study also highlights how OpenCL enables the use of the same codebase on many different systems and hardware platforms without specific code adaptations.</p>
</blockquote>Dr. Hendrik Ranochamail@ranocha.dePhilip Heinisch, Katharina Ostaszewski, and I have published our contribution Towards Green Computing: A Survey of Performance and Energy Efficiency of Different Platforms using OpenCL in the proceedings of the International Workshop on OpenCL IWOCL ‘20. As usual, you can find the preprint and extended information on arXiv.New Paper ‘A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations’ on arXiv2020-06-29T00:00:00+00:002020-06-29T00:00:00+00:00https://ranocha.de/blog/disp_wave_eqs_arXiv<p>Dimitrios Mitsotakis, David I. Ketcheson, and I have published our new paper
<a href="https://arxiv.org/abs/2006.14802">A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations</a>
on arXiv.</p>
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<p>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 for each system. The spatial semidiscretizations are built using the unifying framework of summation by parts operators and include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. Classical time integration schemes such as Runge-Kutta methods and linear multistep methods are applied and the recent relaxation technique is used to enforce temporal conservation.</p>
</blockquote>Dr. Hendrik Ranochamail@ranocha.deDimitrios Mitsotakis, David I. Ketcheson, and I have published our new paper A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations on arXiv.New Paper ‘Positivity-Preserving Adaptive Runge-Kutta Methods’ on arXiv2020-05-14T00:00:00+00:002020-05-14T00:00:00+00:00https://ranocha.de/blog/positive_RK_arXiv<p>Stephan Nüßlein, David I. Ketcheson, and I have published our new paper
<a href="https://arxiv.org/abs/2005.06268">Positivity-Preserving Adaptive Runge-Kutta Methods</a>
on arXiv.</p>
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<p>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. The weights are chosen at each step after calculating the stage derivatives, in a way that also preserves (when possible) the order of accuracy of the method. The choice of weights is given by the solution of a linear program. We investigate different approaches to choosing the weights by considering adding further constraints. We also provide some analysis of the properties of Runge-Kutta methods with perturbed weights. Numerical examples demonstrate the effectiveness of the approach, including application to both stiff and non-stiff problems.</p>
</blockquote>Dr. Hendrik Ranochamail@ranocha.deStephan Nüßlein, David I. Ketcheson, and I have published our new paper Positivity-Preserving Adaptive Runge-Kutta Methods on arXiv.New Paper ‘On Strong Stability of Explicit Runge-Kutta Methods for Nonlinear Semibounded Operators’ published in IMA Journal of Numerical Analysis2020-04-07T00:00:00+00:002020-04-07T00:00:00+00:00https://ranocha.de/blog/SSPRK_stability<p>My paper
<a href="https://dx.doi.org/10.1093/imanum/drz070">On Strong Stability of Explicit Runge-Kutta Methods for Nonlinear Semibounded Operators</a>
has been published in
IMA Journal of Numerical Analysis.
As usual, you can <a href="https://arxiv.org/abs/1811.11601">find the preprint on arXiv</a>.
<a href="https://academic.oup.com/imajna/advance-article/doi/10.1093/imanum/drz070/5760668?guestAccessKey=f584837a-ee73-47d9-b91f-0e38fb9d9bce"> A free access
version of the article is available online</a>.</p>
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<p>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 investigations of high-order methods for hyperbolic conservation laws are often conducted only for the semidiscrete versions. Here, strong stability (also known as monotonicity) of explicit Runge-Kutta methods for ODEs with nonlinear and semibounded (also known as dissipative) operators is investigated. Contrary to the linear case it is proven that many strong-stability-preserving (SSP) schemes of order 2 or greater are not strongly stable for general smooth and semibounded nonlinear operators. Additionally, it is shown that there are first-order-accurate explicit SSP Runge–Kutta methods that are strongly stable (monotone) for semibounded (dissipative) and Lipschitz continuous operators.</p>
</blockquote>Dr. Hendrik Ranochamail@ranocha.deMy paper On Strong Stability of Explicit Runge-Kutta Methods for Nonlinear Semibounded Operators has been published in IMA Journal of Numerical Analysis. As usual, you can find the preprint on arXiv. A free access version of the article is available online.