Multiple-Relaxation Runge Kutta Methods for Conservative Dynamical Systems

We generalize the idea of relaxation time stepping methods in order to preserve multiple nonlinear conserved quantities of a dynamical system by projecting along directions defined by multiple time stepping algorithms. Similar to the directional projection method of Calvo et. al., we use embedded Runge-Kutta methods to facilitate this in a computationally efficient manner. Proof of the accuracy of the modified RK methods and the existence of valid relaxation parameters are given, under some restrictions. Among other examples, we apply this technique to Implicit-Explicit Runge-Kutta time integration for the Korteweg-de Vries equation and investigate the feasibility and effect of conserving multiple invariants for multi-soliton solutions.

[1]  Hendrik Ranocha,et al.  On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations , 2021, Partial Differential Equations and Applications.

[2]  D. Ketcheson,et al.  A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations , 2020, Communications in Computational Physics.

[3]  D. Ketcheson,et al.  General relaxation methods for initial-value problems with application to multistep schemes , 2020, Numerische Mathematik.

[4]  D. Ketcheson,et al.  Relaxation Runge–Kutta Methods for Hamiltonian Problems , 2020, Journal of Scientific Computing.

[5]  Lisandro Dalcin,et al.  Relaxation Runge-Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations , 2019, SIAM J. Sci. Comput..

[6]  M. Uzunca,et al.  Preservation of the invariants of Lotka-Volterra equations by iterated deferred correction methods , 2019, 1901.03870.

[7]  David I. Ketcheson,et al.  Relaxation Runge-Kutta Methods: Conservation and Stability for Inner-Product Norms , 2019, SIAM J. Numer. Anal..

[8]  R. Militaru,et al.  Geometrical Methods and Numerical Computations forPrey-Predator Systems , 2015 .

[9]  Manuel Calvo,et al.  Error growth in the numerical integration of periodic orbits , 2011, Math. Comput. Simul..

[10]  Manuel Calvo,et al.  On the Preservation of Invariants by Explicit Runge-Kutta Methods , 2006, SIAM J. Sci. Comput..

[11]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[12]  M. Carpenter,et al.  Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations , 2003 .

[13]  A. Durán,et al.  The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation , 2000 .

[14]  B. Cano,et al.  Error Growth in the Numerical Integration of Periodic Orbits, with Application to Hamiltonian and Reversible Systems , 1997 .

[15]  J. Sanz-Serna,et al.  Accuracy and conservation properties in numerical integration: the case of the Korteweg-de Vries equation , 1997 .

[16]  L. Vu-Quoc,et al.  Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation , 1995 .

[17]  Mari Paz Calvo,et al.  The Development of Variable-Step Symplectic Integrators, with Application to the Two-Body Problem , 1993, SIAM J. Sci. Comput..

[18]  C. W. Gear,et al.  Invariants and numerical methods for ODEs , 1992 .

[19]  ShuChi-Wang,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes, II , 1989 .

[20]  G. J. Cooper.  Stability of Runge-Kutta Methods for Trajectory Problems , 1987 .

[21]  J. Verwer,et al.  Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations , 1984 .

[22]  J. Dormand,et al.  High order embedded Runge-Kutta formulae , 1981 .

[23]  E. Fehlberg,et al.  Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems , 1969 .

[24]  A. Arakawa.  Computational design for long-term numerical integration of the equations of fluid motion: two-dimen , 1997 .