Abstract
AbstractIn this article, we present a new class of semi‐Lagrangian methods for the numerical approximation of hyperbolic conservation laws. The methods provide a strategy for resolving nonlinear hyperbolic conservations laws within the framework of semi‐Lagrangian discontinuous Galerkin finite‐element methods (SLDG) initially proposed for linear advection problems [J‐M. Qiu and C‐W. Shu, J Comput Phys 230 (2011)]. Using a flux‐correction technique we combine the good numerical properties of the SLDG methods with those of the well‐known Runge‐Kutta discontinuous Galerkin methods (RKDG). The high‐order accuracy and local conservative properties of the resulting schemes make them very attractive for convection‐dominated convection‐diffusion problems, and conservation laws in particular. The semi‐Lagrangian component of the schemes allows for the use of larger Courant numbers than is possible when using high‐order RKDG methods alone.
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