Locally corrected multidimensional quadrature rules for singular functions
SIAM Journal on Scientific Computing
High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions
SIAM Journal on Numerical Analysis
Hybrid Gauss-Trapezoidal Quadrature Rules
SIAM Journal on Scientific Computing
Particle methods for dispersive equations
Journal of Computational Physics
A new version of the fast multipole method for screened Coulomb interactions in three dimensions
Journal of Computational Physics
An adaptive fast solver for the modified Helmholtz equation in two dimensions
Journal of Computational Physics
Multi-symplectic integration of the Camassa-Holm equation
Journal of Computational Physics
Journal of Computational Physics
An operator splitting method for the Degasperis-Procesi equation
Journal of Computational Physics
A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation
Journal of Computational Physics
A sixth-order dual preserving algorithm for the Camassa-Holm equation
Journal of Computational and Applied Mathematics
A self-adaptive moving mesh method for the Camassa-Holm equation
Journal of Computational and Applied Mathematics
Geometric finite difference schemes for the generalized hyperelastic-rod wave equation
Journal of Computational and Applied Mathematics
Convergence of a Particle Method and Global Weak Solutions of a Family of Evolutionary PDEs
SIAM Journal on Numerical Analysis
Hi-index | 31.47 |
An asymptotic higher-order model of wave dynamics in shallow water is examined in a combined analytical and numerical study, with the aim of establishing robust and efficient numerical solution methods. Based on the Hamiltonian structure of the nonlinear equation, an algorithm corresponding to a completely integrable particle lattice is implemented first. Each "particle" in the particle method travels along a characteristic curve. The resulting system of nonlinear ordinary differential equations can have solutions that blow-up in finite time. We isolate the conditions for global existence and prove l1-norm convergence of the method in the limit of zero spatial step size and infinite particles. The numerical results show that this method captures the essence of the solution without using an overly large number of particles. A fast summation algorithm is introduced to evaluate the integrals of the particle method so that the computational cost is reduced from O(N2) to O(N), where N is the number of particles. The method possesses some analogies with point vortex methods for 2D Euler equations. In particular, near singular solutions exist and singularities are prevented from occurring in finite time by mechanisms akin to those in the evolution of vortex patches. The second method is based on integro-differential formulations of the equation. Two different algorithms are proposed, based on different ways of extracting the time derivative of the dependent variable by an appropriately defined inverse operator. The integro-differential formulations reduce the order of spatial derivatives, thereby relaxing the stability constraint and allowing large time steps in an explicit numerical scheme. In addition to the Cauchy problem on the infinite line, we include results on the study of the nonlinear equation posed in the quarter (space-time) plane. We discuss the minimum number of boundary conditions required for solution uniqueness and illustrate this with numerical examples.