Article
Mathematics, Applied
Xueyu Qin, Zhenhua Jiang, Jian Yu, Lintao Huang, Chao Yan
Summary: In this study, explicit strong stability-preserving (SSP) three-derivative Runge-Kutta (ThDRK) methods are proposed and their order accuracy conditions are determined. The SSP theory is developed based on a new Taylor series condition for ThDRK methods, and the optimal SSP coefficient is found. Comparison with other methods shows that ThDRK methods have the highest effective SSP coefficient for order accuracy (3 = p = 5). Numerical experiments demonstrate that ThDRK methods maintain the desired order of convergence and have efficient computational cost.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Benjamin Yeager, Ethan Kubatko, Dylan Wood
Summary: In this work, the linear stability properties of discontinuous Galerkin spatial discretizations with strong-stability-preserving multistep Runge-Kutta methods are assessed. It is found that the constraint for linear stability is more strict than that for strong-stability-preservation. Through testing, an optimal time stepper that requires the fewest evaluations of the discontinuous Galerkin operator is selected, and all methods are found to converge under the stability constraints determined in the study.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
A. Moradi, A. Abdi, G. Hojjati
Summary: This paper discusses implicit-explicit (IMEX) methods for systems of ordinary differential equations. The explicit part of the methods has strong stability preserving (SSP) property, while the implicit part has Runge-Kutta stability property and A- or L-stability. The explicit part is treated by explicit second derivative general linear methods, and the implicit part is treated by implicit general linear methods. Various methods with different orders and combinations of orders are constructed, considering the interaction between the implicit and explicit parts. The performance of the proposed IMEX schemes is tested on one-dimensional linear and nonlinear problems, and the expected order of convergence is presented.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Imre Fekete, Sidafa Conde, John N. Shadid
Summary: We construct a family of embedded pairs for optimal explicit strong stability preserving Runge-Kutta methods, which are used to obtain numerical solutions of spatially discretized hyperbolic PDEs. The new family of embedded pairs offers adaptability through varying the step-size and shows effectiveness in terms of work versus precision, accuracy, and stability.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Michelle Muniz, Matthias Ehrhardt, Michael Guenther, Renate Winkler
Summary: In this paper, numerical methods for solving nonlinear Ito stochastic differential equations on manifolds are presented. The strong convergence of the extended Runge-Kutta-Munthe-Kaas (RKMK) schemes for stochastic ordinary differential equations on manifolds is analysed. The effectiveness of these schemes is demonstrated by numerical results of applying them to a problem with an autonomous underwater vehicle.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics
Giuseppe Izzo, Zdzislaw Jackiewicz
Summary: This paper reviews strong stability preserving discrete variable methods for differential systems and revisits the analysis of strong stability preserving Runge-Kutta methods. Using a new approach, explicit and implicit strong stability preserving Runge-Kutta methods up to the order four are derived, and the strong stability preserving linear multistep methods are also investigated.
BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
Wansheng Wang, Shoufu Li
Summary: Strong stability-preserving time discretization schemes preserve the stability properties of the exact solution and have proved very useful in scientific and engineering computation, especially in solving hyperbolic partial differential equations. The main aim of this work is to further extend this to exponential stability-preserving numerical methods for a general coercive system whose solution is exponentially growing or decaying. The new developments in this paper also include their applications to various linear and nonlinear evolution problems.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Juntao Huang, Thomas Izgin, Stefan Kopecz, Andreas Meister, Chi-Wang Shu
Summary: In this paper, a stability analysis is performed for strong-stability-preserving modified Patankar-Runge-Kutta (SSPMPRK) schemes, which can be used to solve convection equations with stiff source terms. The analysis identifies the range of free parameters in these schemes for stability. Numerical experiments validate the analysis.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Pari Khakzad, Afsaneh Moradi, Gholamreza Hojjati, Mohammad Mehdizadeh Khalsaraei, Ali Shokri
Summary: This paper discusses numerical methods for solving systems of differential equations with both linear and nonlinear components arising from the semi-discretization of certain partial differential equations. Strong stability preserving (SSP) methods are introduced to overcome the convergence limitations of linear stability properties when dealing with problems with discontinuous solutions. The integrating factor approach is used to reduce the constraints on time-stepping in SSP implicit-explicit methods by solving the linear part exactly. The paper presents the construction and sufficient conditions for integrating factor general linear methods (IFGLMs) with strong stability properties and verifies the results numerically on representative test cases.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Rachid Ait-Haddou
Summary: This paper discusses the computation of optimal linear SSP coefficients for explicit one-step methods, providing upper and lower bounds based on generalized Laguerre orthogonal polynomials. An efficient algorithm for computing these coefficients and their associated optimal linear SSP methods is proposed, along with adaptive spectral transformations of measures. The complexity of the algorithm depends only on the order of the method, independent of the number of stages.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
T. Dzanic, W. Trojak, F. D. Witherden
Summary: In this work, a modified explicit Runge-Kutta temporal integration scheme is proposed to guarantee the preservation of locally-defined quasiconvex set of bounds for the solution. The schemes use a bijective mapping to enforce bounds between the admissible set of solutions and the real domain. It is shown that these schemes can recover a wide range of methods, including positivity preserving, discrete maximum principle satisfying, entropy dissipative, and invariant domain preserving schemes. The additional computational cost is the evaluation of two nonlinear mappings which generally have closed-form solutions. The approach is demonstrated in numerical experiments using a pseudospectral spatial discretization without explicit shock capturing schemes for nonlinear hyperbolic problems with discontinuities.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Gui-Lai Zhang
Summary: This paper defines impulsive one-step numerical methods, proves their convergence and order, introduces another equivalent form, and validates their advantages through numerical experiments.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Chaolong Jiang, Yushun Wang, Yuezheng Gong
Summary: A novel class of explicit high-order energy-preserving methods for general Hamiltonian partial differential equations with non-canonical structure matrix is proposed. By reformulating the original system into an equivalent form with a modified quadratic energy conservation law and discretizing it in time using explicit high-order Runge-Kutta methods with orthogonal projection techniques, the schemes are shown to share the order of explicit Runge-Kutta method and preserve energy while reaching high-order accuracy.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Stephan Nuesslein, Hendrik Ranocha, David Ketcheson
Summary: This paper proposes a method to ensure the solution of differential equations remains positive or within a certain range by adjusting the weights of Runge-Kutta integration. The weights are chosen by solving a linear program and further constraints are considered for selecting the weights. Numerical examples demonstrate the effectiveness of this approach in tackling both stiff and non-stiff problems.
COMMUNICATIONS IN APPLIED MATHEMATICS AND COMPUTATIONAL SCIENCE
(2021)
Article
Mathematics, Applied
Zhaohui Fu, Tao Tang, Jiang Yang
Summary: This study aims to extend the strong stability preserving (SSP) theory to solve the nonlinear phase field equation that satisfies both the maximum bound property (MBP) and the energy dissipation law. By using the Runge-Kutta time discretization method, we derive a necessary and sufficient condition for satisfying MBP, and further provide a necessary condition for the MBP scheme to satisfy energy dissipation.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
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
Michael Schlottke-Lakemper, Andrew R. Winters, Hendrik Ranocha, Gregor J. Gassner
Summary: The method reformulates the elliptic problem into a hyperbolic diffusion problem to compute gravitational forces in self-gravitating flows, using a high-order discontinuous Galerkin method in pseudotime. The flow and gravity solvers run on a joint hierarchical Cartesian mesh and are two-way coupled via source terms. A key advantage is the reuse of existing explicit hyperbolic solvers while retaining advanced features like non-conforming and solution-adaptive grids.
JOURNAL OF 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)
Article
Astronomy & Astrophysics
Katharina Ostaszewski, Karl-Heinz Glassmeier, Charlotte Goetz, Philip Heinisch, Pierre Henri, Sang A. Park, Hendrik Ranocha, Ingo Richter, Martin Rubin, Bruce Tsurutani
Summary: The study presents 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. It identifies these enhancements as magnetosonic waves and uses machine learning to identify around 70,000 steepened waves. The occurrence of these waves is linked to the activity of the comet, primarily observed at high outgassing rates.
ANNALES GEOPHYSICAE
(2021)
