Semi-Lagrangian one-step methods for two classes of time-dependent partial differential systems

Semi-Lagrangian methods are numerical methods designed to find approximate solutions to particular time-dependent partial differential equations (PDEs) that describe the advection process. We propose semi-Lagrangian one-step methods for numerically solving initial value problems for two general systems of partial differential equations. Along the characteristic lines of the PDEs, we use ordinary differential equation (ODE) numerical methods to solve the PDEs. The main benefit of our methods is the efficient achievement of high order local truncation error through the use of Runge-Kutta methods along the characteristics. In addition, we investigate the numerical analysis of semi-Lagrangian methods applied to systems of PDEs: stability, convergence, and maximum error bounds.