The Spatial-Temporal Fourth-Order Conservative Characteristic Runge-Kutta Finite Difference Method for Convection-Dominated Diffusion Equation (Advances in Applied Mathematics and Mechanics, 2025)
In this paper, we develop a new class of conservative characteristic finitedifference methods for solving convection-dominated diffusion problems with fourth-order accuracy in both time and space. Specifically, the method of characteristics is utilized to handle the convection term, which allows for greater flexibility in the choice oftime step sizes. To achieve high-order temporal accuracy, we propose characteristics-based optimal implicit strong stability preserving (SSP) Runge-Kutta methods implemented along the streamline. Furthermore, a conservative interpolation is employedto calculate values at the tracking points. By introducing diverse fourth-order approximation operators on the uniform Eulerian and irregular Lagrangian meshes, we candeal with the diffusion term with high accuracy while preserving the conservationproperty. The mass conservation for our proposed method is theoretically proved, andis verified through numerical experiments. Moreover, the numerical tests demonstratethat our scheme achieves temporal and spatial fourth-order accuracy and generatesnon-oscillatory solutions, even with large time step sizes.