Sympletic Finite Difference Approximations of the Nonlinear Klein--Gordon Equation

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

“…The nonlinear Klein-Gordon equation with cubic nonlinearity [47][48][49][50] the absolute error for energy function is given by…”

The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

“…The investigation on symplectic integration for the equation (3.1) can be found in [4] and the references therein. The case (4) in Theorem 2.1 is suitable for the equation (3.2).…”

“…A systemic theory of symplectic integrators for Hamiltonian ODEs has been established by some authors. The Runge-Kutta methods play an important role in numerically solving differential equations (see [1], [3], [4], [6]- [16] and the references therein). The symplectic condition of Runge-Kutta methods was found independently by Lasagni, Sanz-Serna and Suris in 1988 (see [10], [12], [15] and the references therein).…”

“…The symplectic condition of Runge-Kutta methods was found independently by Lasagni, Sanz-Serna and Suris in 1988 (see [10], [12], [15] and the references therein). The numerical analysis has been investigated and developed by some authors (see [4], [6], [7], [10]- [16] and the references therein). Some characterizations of symplectic partitioned Runge-Kutta methods, which are very useful for the construction of symplectic schemes for solving numerical Hamiltonian problems, were obtained by Sanz-Serna in [10], Sun in [13], [14] and Suris in [16] and recently discussed by Marsden and West in [8].…”

Finite difference solution of a nonlinear Klein-Gordon equation with an external source

“…Certain numerical methods (the symplectic and multisymplectic finite difference schemes and the spectral methods, etc.) are devoted to homogeneous nonlinear Klein-Gordon equations [26]- [30]. In [30] the authors propose a spectral method using as a basis the Legendre functions.…”