Article
Mathematics, Applied
Johnathon Upperman, Nail K. K. Yamaleev
Summary: In this paper, a positivity-preserving, entropy stable first-order scheme for the three-dimensional compressible Navier-Stokes equations is developed based on a one-dimensional scheme. The new scheme is proven to be entropy stable, design-order accurate for smooth solutions, and guarantees the pointwise positivity of thermodynamic variables for 3-D compressible viscous flows.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Jan Nordstrom
Summary: We investigate a specific skew-symmetric form of nonlinear hyperbolic problems and find that it leads to bounds on energy and entropy. Taking the compressible Euler equations as an example, we transform them into skew-symmetric form and obtain estimates for energy and entropy. Finally, we demonstrate that the skew-symmetric formulation allows for energy and entropy stable discrete approximations if the scheme is formulated on a summation-by-parts form.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Nail K. Yamaleev, Johnathon Upperman
Summary: This paper extends the high-order positivity-preserving, entropy stable spectral collocation schemes developed for the one-dimensional compressible Navier-Stokes equations to three spatial dimensions. The proposed schemes combine a positivity-violating entropy stable method with a novel first-order positivity-preserving entropy stable finite volume-type scheme. The schemes achieve positivity-preserving and excellent discontinuity-capturing properties by adding artificial dissipation in the form of low- and high-order Brenner-Navier-Stokes diffusion operators. The new schemes are also entropy conservative and freestream preserving.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Magnus Svard
Summary: This study focuses on deriving boundary conditions for the initial-boundary value Euler equations to establish an entropy bound for the physical (Navier-Stokes) entropy. The research begins by reviewing the entropy bound obtained for standard no-penetration wall boundary conditions and proposes a numerical implementation. The main results include deriving full-state boundary conditions and demonstrating that linear theory boundary conditions are unable to bound the entropy, requiring nonlinear bounds and additional boundary conditions. The theoretical findings are supported by numerical experiments.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Johnathon Upperman, Nail K. Yamaleev
Summary: This paper introduces a new class of positivity-preserving, entropy stable schemes for the 1-D compressible Navier-Stokes equations. The proposed scheme satisfies the discrete entropy inequality and ensures the pointwise positivity of density, temperature, pressure, and internal energy. It uses high-order spectral collocation methods and artificial dissipation to regularize the Navier-Stokes equations and eliminate time step stiffness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Nail K. Yamaleev, Johnathon Upperman
Summary: This paper presents a new positivity-preserving, entropy stable spectral collocation scheme for the 1-D compressible Navier-Stokes equations. The method guarantees the pointwise positivity of thermodynamic variables and captures discontinuities effectively.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Remi Abgrall, Philipp Oeffner, Hendrik Ranocha
Summary: This paper proposes an approach to construct entropy conservative/dissipative semidiscretizations in the general class of residual distribution (RD) schemes. The approach involves adding suitable correction terms characterized as solutions of certain optimization problems. The method is applied to the SBP- SAT framework and novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed. Explicit solutions are provided for all optimization problems, and a fully discrete entropy conservative/dissipative RD scheme is obtained using the deferred correction method for time integration.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Jan Nordstrom, Fredrik Lauren
Summary: Energy stable and conservative nonlinear weakly imposed interface conditions for the incompressible Euler equations are derived in the continuous setting. The numerical scheme is proved to be stable and conservative by discretely mimicking the continuous analysis using summation-by-parts operators. The theoretical findings are verified by numerical experiments.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Computer Science, Interdisciplinary Applications
Ayoub Gouasmi, Scott M. Murman, Karthik Duraisamy
Summary: Entropy-Stable (ES) schemes have gained increasing interest in the past decade, particularly in the context of under-resolved simulations of compressible turbulent flows. This study investigates the accuracy degradation issues of ES schemes in the low-Mach-number regime and explores the use of Flux-Preconditioning to improve their behavior without compromising entropy-stability.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Carmela Scalone
Summary: This paper discusses the ability of stochastic theta-Milstein methods to generate positive sequences of numerical approximations in models with positive solutions. It also introduces a class of theta-methods that can preserve the positivity of a jump extended nonlinear CIR model, with numerical experiments confirming the theoretical analysis.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Engineering, Multidisciplinary
Guosheng Fu, Zhiliang Xu
Summary: We introduce a novel class of high-order space-time finite element schemes for solving the Poisson-Nernst-Planck (PNP) equations. Our schemes achieve mass conservation, positivity preservation, and unconditional energy stability for any order of approximation. This is accomplished by employing the entropy variable formulation and a discontinuous Galerkin discretization in time.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Philippe G. LeFloch, Hendrik Ranocha
Summary: This study investigates numerical methods for nonlinear hyperbolic conservation laws with non-convex flux, computing kinetic functions to characterize macro-scale dynamics. It demonstrates that entropy stability does not guarantee uniqueness of numerical solutions, and designs entropy-dissipative schemes for systems with delta shocks.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Tristan Montoya, David W. Zingg
Summary: In this paper, we propose a unified framework for the matrix-based formulation and analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods for conservation laws on general unstructured grids. We use the multidimensional summation-by-parts (SBP) property to establish the discrete equivalence of strong and weak formulations, as well as the conservation and energy stability properties. Numerical experiments are conducted to validate the theoretical analysis.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Xinhui Wu, Ethan J. Kubatko, Jesse Chan
Summary: The study introduces a high-order entropy stable discontinuous Galerkin method for two dimensional shallow water equations on curved triangular meshes, maintaining a semi-discrete entropy inequality and balance for continuous bathymetry profiles. Numerical experiments confirm the high-order accuracy and theoretical properties of the scheme, comparing it to other entropy stable schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Hendrik Ranocha, Gregor J. Gassner
Summary: The study investigates the local linear stability issues of entropy-conserving/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations and examines the impact of pressure equilibrium preservation on these issues.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Hendrik Ranocha
Summary: The study investigates the strong stability of explicit Runge-Kutta methods for ODEs with nonlinear and semibounded operators, proving that many second-order or higher SSP schemes are not strongly stable for general smooth and semibounded nonlinear operators. It is also shown that there are first-order accurate explicit SSP Runge-Kutta methods that are strongly stable for semibounded and Lipschitz continuous operators.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Computer Science, Interdisciplinary Applications
Diego Rojas, Radouan Boukharfane, Lisandro Dalcin, David C. Del Rey Fernandez, Hendrik Ranocha, David E. Keyes, Matteo Parsani
Summary: In computational fluid dynamics, transformational advances in individual components of future solvers are needed to meet the demand for reliable simulations in increasingly multidisciplinary fields. While hardware compatibility and efficiency are crucial, algorithmic robustness with minimal user intervention is equally important for viability.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Hendrik Ranocha, Jan Nordstrom
Summary: This article proposes a novel class of A stable SBP time integration methods to investigate the property of summation by parts in numerical methods, using a projection method to impose initial conditions strongly without compromising stability. The new methods also include the classical Lobatto IIIA collocation method.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Philippe G. LeFloch, Hendrik Ranocha
Summary: This study investigates numerical methods for nonlinear hyperbolic conservation laws with non-convex flux, computing kinetic functions to characterize macro-scale dynamics. It demonstrates that entropy stability does not guarantee uniqueness of numerical solutions, and designs entropy-dissipative schemes for systems with delta shocks.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Physics, Mathematical
Hendrik Ranocha, Dimitrios Mitsotakis, David Ketcheson
Summary: The study introduces a general framework for designing conservative numerical methods, integrating summation by parts operators, split forms, and relaxation Runge-Kutta methods. This framework is applied to create new classes of fully-discrete conservative methods for nonlinear dispersive wave equations, demonstrating their favorable properties through numerical tests.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Remi Abgrall, Philipp Oeffner, Hendrik Ranocha
Summary: This paper proposes an approach to construct entropy conservative/dissipative semidiscretizations in the general class of residual distribution (RD) schemes. The approach involves adding suitable correction terms characterized as solutions of certain optimization problems. The method is applied to the SBP- SAT framework and novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed. Explicit solutions are provided for all optimization problems, and a fully discrete entropy conservative/dissipative RD scheme is obtained using the deferred correction method for time integration.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Hendrik Ranocha
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Physics, Multidisciplinary
Jesse Chan, Hendrik Ranocha, Andres M. Rueda-Ramirez, Gregor Gassner, Tim Warburton
Summary: 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 Discontinuous Galerkin (DG) methods which incorporate an entropy projection are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.
FRONTIERS IN PHYSICS
(2022)
Article
Computer Science, Software Engineering
David I. Ketcheson, Hendrik Ranocha
Summary: We introduce BSeries.jl, a Julia package that facilitates computation and manipulation of B-series. B-series are versatile theoretical tools used to understand and design discretizations of differential equations. We provide an overview of B-series theory and associated concepts, and offer examples of their application, including method composition and backward error analysis. The software is highly efficient and capable of working with high-order B-series.
ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
(2023)
Article
Mathematics, Applied
Davide Torlo, Philipp Oeffner, Hendrik Ranocha
Summary: This article discusses the methods to analyze the performance and robustness of Patankar-type schemes, and demonstrates their problematic behavior on both linear and nonlinear stiff problems.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Hendrik Ranocha, Andrew R. Winters, Hugo Guillermo Castro, Lisandro Dalcin, Michael Schlottke-Lakemper, Gregor J. Gassner, Matteo Parsani
Summary: We study the temporal step size control of explicit Runge-Kutta methods for compressible computational fluid dynamics. We compare error-based approaches to classical step size control based on a CFL number and demonstrate that error-based methods are more convenient and efficient in various applications. Our numerical examples show the effectiveness of error-based step size control in different scenarios.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Computer Science, Software Engineering
Viktor Linders, Hendrik Ranocha, Philipp Birken
Summary: This article investigates the entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations, and explores the influence of iterative methods on solving the resulting nonlinear equations. It is found that Newton's method can turn an entropy dissipative scheme into an anti-dissipative one, even with smaller iteration errors than time integration errors. A relaxation technique is proposed as the most effective remedy, which is originally designed to fix entropy errors in time integration methods. The research findings are corroborated by considering Burgers' equation and nonlinear dispersive wave equations, showing that entropy conservation leads to more accurate numerical solutions compared to non-conservative schemes, even with a tolerance order of magnitude larger.
BIT NUMERICAL MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Hendrik Ranocha
Summary: Nishikawa (2007) proposed a reformulation of the classical Poisson equation as a steady state problem for a linear hyperbolic system, which provides optimal error estimates for the solution of the elliptic equation and its gradient. However, it hinders the use of well-known solvers for elliptic problems. We establish connections to a discontinuous Galerkin (DG) method studied by Cockburn, Guzman, and Wang (2009) that is generally difficult to implement. Additionally, we demonstrate the efficient implementation of this method using summation by parts (SBP) operators, particularly in the context of SBP DG methods like the DG spectral element method (DGSEM). The resulting scheme combines desirable properties from both the hyperbolic and the elliptic perspective, offering a higher order of convergence for the gradients than what is typically expected from DG methods for elliptic problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Hendrik Ranocha, Lisandro Dalcin, Matteo Parsani, David Ketcheson
Summary: The study developed error-control based time integration algorithms for compressible fluid dynamics (CFD) applications, demonstrating their efficiency and robustness. Focusing on discontinuous spectral element semidiscretizations, new controllers were designed for existing methods and some new embedded Runge-Kutta pairs. By comparing error-control based methods with the common CFL number approach, optimized methods showed improved performance and adaptability.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Hendrik Ranocha, Gregor J. Gassner
Summary: The study investigates the local linear stability issues of entropy-conserving/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations and examines the impact of pressure equilibrium preservation on these issues.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
