Relaxation Runge-Kutta methods for Hamiltonian problems

DOI10.1007/s10915-020-01277-yzbMath1456.65050arXiv2001.04826OpenAlexW3103244101MaRDI QIDQ777041

David I. Ketcheson, Hendrik Ranocha

Publication date: 13 July 2020

Published in: Journal of Scientific Computing (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2001.04826

zbMATH Keywords

energy conservationRunge-Kutta methodsHamiltonian problemsstructure preservationgeometric numerical integration

Mathematics Subject Classification ID

Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) Stability and convergence of numerical methods for ordinary differential equations (65L20) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs (65M70) Numerical methods for Hamiltonian systems including symplectic integrators (65P10)

Related Items

Implicit-explicit relaxation Runge-Kutta methods: construction, analysis and applications to PDEs, Entropy-preserving and entropy-stable relaxation IMEX and multirate time-stepping methods, Geometric Integration of ODEs Using Multiple Quadratic Auxiliary Variables, Relaxation Exponential Rosenbrock-Type Methods for Oscillatory Hamiltonian Systems, Resolving entropy growth from iterative methods, Linearly implicit and high-order energy-preserving relaxation schemes for highly oscillatory Hamiltonian systems, Fully discrete explicit locally entropy-stable schemes for the compressible Euler and Navier-Stokes equations, Multiple-relaxation Runge Kutta methods for conservative dynamical systems, General relaxation methods for initial-value problems with application to multistep schemes, Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation, A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations, A new class of \(A\) stable summation by parts time integration schemes with strong initial conditions, Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts, Energy Stability of Explicit Runge--Kutta Methods for Nonautonomous or Nonlinear Problems, On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations, Positivity-preserving adaptive Runge-Kutta methods, A Conservative Fully Discrete Numerical Method for the Regularized Shallow Water Wave Equations, Relaxation deferred correction methods and their applications to residual distribution schemes, Symplectic central difference scheme for quasi-linear autonomous Birkhoffian systems, On Energy Laws and Stability of Runge--Kutta Methods for Linear Seminegative Problems

Uses Software

• Mathematica
• RODAS
• Qhull
• SciPy
• Python
• GitHub
• RRK_rr

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Error growth in the numerical integration of periodic orbits
• Projection methods preserving Lyapunov functions
• A method for the integration in time of certain partial differential equations
• Efficient implementation of essentially nonoscillatory shock-capturing schemes
• High order embedded Runge-Kutta formulae
• An explicit finite-difference scheme with exact conservation properties
• Accuracy and conservation properties in numerical integration: The case of the Korteweg-de Vries equation
• Energy behaviour of the Boris method for charged-particle dynamics
• An efficient Runge-Kutta \((4,5)\) pair
• Preserving algebraic invariants with Runge-Kutta methods
• Lack of dissipativity is not symplecticness
• Mimetic properties of difference operators: product and chain rules as for functions of bounded variation and entropy stability of second derivatives
• Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations
• Time integration and discrete Hamiltonian systems
• Stability of Runge-Kutta Methods for Trajectory Problems
• The Development of Variable-Step Symplectic Integrators, with Application to the Two-Body Problem
• On the Preservation of Invariants by Explicit Runge--Kutta Methods
• Geometric integration using discrete gradients
• The quickhull algorithm for convex hulls
• Error Growth in the Numerical Integration of Periodic Orbits, with Application to Hamiltonian and Reversible Systems
• On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators
• Solving Ordinary Differential Equations II
• On energy conservation of the simplified Takahashi-Imada method
• Relaxation Runge--Kutta Methods: Conservation and Stability for Inner-Product Norms
• Relaxation Runge--Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier--Stokes Equations
• A new class of energy-preserving numerical integration methods
• Geometric Numerical Integration
• Numerical Methods for Ordinary Differential Equations
• A comparison of symplectic and Hamilton's principle algorithms for autonomous and non-autonomous systems of ordinary differential equations