General Relaxation Methods for Initial-Value Problems with Application to Multistep Schemes

Abstract

Recently, an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial-value problems, using Runge-Kutta methods. We generalize this approach to multistep methods, including all general linear methods of order two or higher, and many other classes of schemes. We prove the existence of a valid relaxation parameter and high-order accuracy of the resulting method, in the context of general equations, including but not limited to conservative or dissipative systems. The theory is illustrated with several numerical examples.

References in corpus (12)

• Review of Summation-by-parts schemes for initial-boundary-value problems
• On discretely entropy conservative and entropy stable discontinuous Galerkin methods
• Relaxation Runge-Kutta Methods: Fully-Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations
• Comparison of some Entropy Conservative Numerical Fluxes for the Euler Equations
• High-Order Implicit Time-Marching Methods Based on Generalized Summation-By-Parts Operators
• Relaxation Runge-Kutta Methods for Hamiltonian Problems
• Fully-Discrete Explicit Locally Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations
• Generalised Summation-by-Parts Operators and Variable Coefficients
• On Strong Stability of Explicit Runge-Kutta Methods for Nonlinear Semibounded Operators
• Some Notes on Summation by Parts Time Integration Methods
• On the zero-stability of multistep methods on smooth nonuniform grids
• Entropy production by implicit Runge-Kutta schemes

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• A conservative fully-discrete numerical method for the regularised shallow water wave equations
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• Traveling-wave solutions and structure-preserving numerical methods for a hyperbolic approximation of the Korteweg-de Vries equation
• Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts
• On the Rate of Error Growth in Time for Numerical Solutions of Nonlinear Dispersive Wave Equations
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• A New Class of $A$ Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions
• Quasi-Orthogonal Runge-Kutta Projection Methods
• Functional-preserving predictor-corrector multiderivative schemes
• Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics
• Multiderivative time integration methods preserving nonlinear functionals via relaxation
• Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations
• Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions
• Pseudo-Energy-Preserving Explicit Runge-Kutta Methods