A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation

On the development of a high-order compact scheme for exhibiting the switching and dissipative solution natures in the Camassa–Holm equation

“…The local discontinuous Galerkin method was also applied to solve the CH equation with success [6]. Other numerical methods, known as the multi-symplectic method [8], particle method [22][23][24], energy-conserving Galerkin method [29], self-adaptive mesh method [30] and minimized dispersion-error method [31], have also been proposed to solve the CH equation. To get a better predicted result for the soliton-cuspon or the cuspon-cuspon interaction problem, integrable discretizations of the soliton equation have been recently proposed.…”

“…The momentum variable given below is adopted in the so-called u À m formulation of CH equation [31,33] …”

Viscous and inviscid regularizations in a class of evolutionary partial differential equations

“…In particular, if the equation of interest is a dispersive equation, such as equation (1.2), a dispersion-relation-preserving scheme is necessary to ensure the accuracy of numerical solutions. The first-order derivative terms m n+1/2 x and u n+1/2 x are approximated by a sixth-order dispersion-relation-preserving scheme developed by Chiu et al [7] using the nodal values of (2.3). It is worth noting that the midpoint time integrator is a symplectic integrator [4,8], which retains the Hamiltonian invariants of the PDEs, if the spatially discretized PDEs are Hamiltonian systems.…”

“…We remark that in the paper by Chiu et al [7], a two-step iterative algorithm (with the function F in the above algorithm as a special case) was employed to solve the shallow-water wave equation (1.2), a member of the class of PDEs considered in this study. Detailed error analysis for the dispersion-relation-preserving approximation as well as numerical convergence study was carried out in that context, thanks to the availability of exact solutions.…”

An optimized dispersion–relation-preserving combined compact difference scheme to solve advection equations

“…For this purpose, dispersion-relation-preserving (DRP) approaches have been developed to enhance convective stability by rigorously preserving the dispersion relation [1][2][3][4][5][6]. Furthermore, compact difference schemes offer spectral accuracy with fewer grid points to improve convective stability [7][8][9][10].…”