Pseudospectral Method for the "Good" Boussinesq Equation

Efficient structure‐preserving schemes for good Boussinesq equation

“…The nonlinear stability and the convergence of FDM for GB equation are discussed. () Then, some high accuracy methods were discovered, such as high‐order compact method, spectral as well as pseudo‐spectral methods with exponential convergent rate, energy‐preserving method,() and symplectic structure‐preserving method. ()…”

A Fourier pseudospectral method for the “good” Boussinesq equation with second‐order temporal accuracy

“…Ortega and Sanz‐Serna discussed nonlinear stability and convergence of some simple finite difference schemes for the numerical solution of this equation. More analytical and numerical works related to GB equations can be found in the literature, for example, .…”

“…Many interesting theoretical analysis and numerical results have been reported in the existing literature; for example, see for the semidiscrete spectral methods, for the error estimate of a fully discrete scheme, and for the error estimates of the Benjamin–Ono equation or related nonlocal models, and so forth. For the GB equation , it is worth mentioning De Frutos et al's work on the nonlinear analysis of a second‐order (in time) pseudospectral scheme for the GB equation (with p = 2). However, as the authors point out in their remark on page 119, these theoretical results were not optimal: “… our energy norm is an L 2 ‐norm of u combined with a negative norm of u t .…”

“…The presence of a second‐order spatial derivative for the nonlinear term leads to an essential difficulty of numerical error estimate in a higher order Sobolev norm. In addition to the lack of optimal numerical error estimate, the analysis in also imposes a severe time step restriction: (with C a fixed constant), in the nonlinear stability analysis. Such a constraint becomes very restrictive for a fine numerical mesh and leads to a high computational cost.…”

On Numerical Methods for Hamiltonian PDEs and a Collocation Method for the Vlasov–Maxwell Equations

“…there are boundary terms that arise from integration by This approach, based on discretization of the inner prodparts in the true bracket uct and I operator and approximation of the exact Hamiltonian by a quadrature formula, actually underlies the method used by de Frutos et al [16] for the good Bous-…”