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

## TL;DR

This paper introduces semi-Lagrangian one-step methods utilizing Runge-Kutta schemes for efficiently solving time-dependent PDE systems, with analysis of stability, convergence, and error bounds.

## Contribution

It develops a novel semi-Lagrangian approach for systems of PDEs, combining ODE solvers along characteristics with high-order accuracy and rigorous numerical analysis.

## Key findings

- Achieves high order local truncation error using Runge-Kutta methods
- Provides stability and convergence analysis for the proposed methods
- Establishes maximum error bounds for the numerical solutions

## Abstract

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.