On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D

In this article, one-step semi-Lagrangian method is investigated for computing the numerical solutions of the first-order time-dependent nonlinear partial differential equations in 1D with initial and boundary conditions. This method is based on Lagrangian trajectory or the integration from the departure points to the arrival points (regular nodes) and Runge-Kutta method for ordinary differential equations. The departure points are traced back from the arrival points along the trajectory of the path. The convergence and stability are studied for the implicit and explicit methods. The numerical examples show that those methods work very efficient for the time-dependent nonlinear partial differential equations.