A Lawson-type exponential integrator for the Korteweg–de Vries equation

We propose an explicit numerical method for the periodic Korteweg–de Vries equation. Our method is based on a Lawson-type exponential integrator for time integration and the Rusanov scheme for Burgers’ nonlinearity. We prove first-order convergence in both space and time under a mild Courant–Friedrichs–Lewy condition $\tau =O(h)$, where $\tau$ and $h$ represent the time step and mesh size for solutions in the Sobolev space $H^3((-\pi , \pi ))$, respectively. Numerical examples illustrating our convergence result are given.

NUMERICAL SOLUTION OF A MODEL FOR TURBULENT DIFFUSION

A model is considered for turbulent diffusion which consists of a Riesz space fractional derivative to describe the turbulent phenomenon and also includes advection and classical diffusion. We present a first order explicit numerical method and a second order implicit numerical method to solve our problem and prove convergence results for both methods, including the derivation of stability constraints needed for the explicit numerical method to converge. In the end, to give some insights into the phenomenon of turbulent diffusion described by the Riesz fractional derivative, we show the behavior of the solution when we consider a Gaussian initial condition.

An Explicit Numerical Method for the Fractional Cable Equation

An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula. The accuracy, stability, and convergence of the method are considered. The stability analysis is carried out by means of a kind of von Neumann method adapted to fractional equations. The convergence analysis is accomplished with a similar procedure. The von-Neumann stability analysis predicted very accurately the conditions under which the present explicit method is stable. This was thoroughly checked by means of extensive numerical integrations.