Spectrally accurate space-time solution of Hamiltonian PDEs

Abstract

Recently, the numerical solution of multi-frequency, highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. When the problem derives from the space semi- discretization of (possibly Hamiltonian) partial differential equations (PDEs), the resulting problem may be stiffly-oscillatory, rather than highly-oscillatory. In such a case, a different implementation of the methods is needed, in order to gain the maximum efficiency.

arXiv comment17 pages, 3 figures

References in corpus (4)

• A unifying framework for the derivation and analysis of effective classes of one-step methods for ODEs
• A note on the efficient implementation of Hamiltonian BVMs
• Energy conservation issues in the numerical solution of the semilinear wave equation
• On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems

Cited by in corpus (14)

• Arbitrarily high-order energy-preserving methods for simulating the gyrocenter dynamics of charged particles
• Analysis of Spectral Hamiltonian Boundary Value Methods (SHBVMs) for the numerical solution of ODE problems
• Line Integral solution of Hamiltonian PDEs
• A new framework for polynomial approximation to differential equations
• A spectrally accurate step-by-step method for the numerical solution of fractional differential equations
• A note on the continuous-stage Runge-Kutta-(Nyström) formulation of Hamiltonian Boundary Value Methods (HBVMs)
• Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs
• Arbitrarily high-order energy-conserving methods for Poisson problems
• (Spectral) Chebyshev collocation methods for solving differential equations
• Spectrally accurate space-time solution of Manakov systems
• Numerical solution of FDE-IVPs by using Fractional HBVMs: the fhbvm code
• Solving FDE-IVPs by using Fractional HBVMs: some experiments with the fhbvm code
• A shooting-Newton procedure for solving fractional terminal value problems
• Continuous-Stage Runge-Kutta approximation to Differential Problems