2021 UNC Greensboro PDE Conference. Electron. J. Diff. Eqns., Conference 26 (2022), pp. 97-113.

Semi-Lagrangian forward methods for some time-dependent nonlinear partial differential equations

Daniel X. Guo

In this article, we study one-step Semi-Lagrangian forward method for computing the numerical solutions of time-dependent nonlinear partial differential equations with initial and boundary conditions in one space dimension. Comparing with classic Semi-Lagrangian method, this method is more straight forward to analyze and implement. This method is based on Lagrangian trajectory from the departure points (regular nodes) to the arrival points by Runge-Kutta methods. The arrival points are traced forward from the departure points along the trajectory of the path. Most likely the arrival points are not on the regular grid nodes. However, it is convenient to approximate the high order derivative terms in spatial dimension on regular nodes. The convergence and stability are studied for the explicit methods. The numerical examples show that those methods work very efficient for the time-dependent nonlinear partial differential equations.

Published August 25, 2022.
Math Subject Classifications: 35Q35, 65M12, 76M20.
Key Words: Semi-Lagrangian forward methods; trajectory; Runge-Kutta method; time-dependent nonlinear partial differential equations.

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Daniel Guo
Department of Mathematics and Statistics
University of North Carolina at Wilmington
Wilmington, NC 28403, USA
email: guod@uncw.edu

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