Conservative finite difference schemes for the chiral nonlinear Schrödinger equation

Conservative finite difference schemes for the Degasperis-Procesi equation

We consider the numerical integration of the Degasperis–Procesi equation, which was recently introduced as a completely integrable shallow water equation. For the equation, we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi-Hamiltonian form of the equation at a same time. We also prove the unique solvability of the schemes, and show som...

Numerical Stability of Explicit Runge-Kutta Finite Difference Schemes for the Nonlinear Schrödinger Equation

Article history: Received 8 July 2012 Received in revised form 1 January 2013 Accepted 26 April 2013 Available online 3 May 2013

Nonstandard finite difference schemes for differential equations

In this paper, the reorganization of the denominator of the discrete derivative and nonlocal approximation of nonlinear terms are used in the design of nonstandard finite difference schemes (NSFDs). Numerical examples confirming then efficiency of schemes, for some differential equations are provided. In order to illustrate the accuracy of the new NSFDs, the numerical results are compared with ...

Finite-Difference Schemes for the Diffusion Equation

Abst rac t . The Crank-Nicolson scheme is widely used to solve numerically the diffusion equation, because of its good stability properties. It is, however, ill-behaved when large time-steps are used: the short wave-lengths may happen to be less damped than the long ones. A detailed analysis of this flaw is performed and an Mternative scheme is proposed, which removes this difficulty while pres...

Two Conservative Difference Schemes for the Generalized Rosenau Equation