Implicit-explicit Runge-Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations.

*(English)*Zbl 1281.65123The transient advection-diffusion equation with Dirichlet boundary condition on a certain time interval and polyhedron in \(\mathbb R^d\) for the space variable is investigated from a numerical analysis point of view.

For the discretization, the authors use a two-stage implicit-explicit Runge-Kutta method which is used for the advection-diffusion equation for the first time. The space discretization is performed using continuous, piecewise affine finite elements with continuous interior penalty on the interelement normal gradient jumps as a specific example of symmetric stabilization. The advection and stabilization operators are treated explicitly and the diffusion operator implicitly.

The main result of the paper is to obtain stability and error estimates for smooth solutions. The results are formulated in terms of the Courant and Péclet numbers. Stability and convergence are proved for both cases of advection-dominated and diffusion-dominated regimes.

Finally, two numerical experiments using FreeFem++ are presented to illustrate the analysis, namely, convergence rates to a known smooth solution and control spurious oscillations for a solution with sharp layers.

For the discretization, the authors use a two-stage implicit-explicit Runge-Kutta method which is used for the advection-diffusion equation for the first time. The space discretization is performed using continuous, piecewise affine finite elements with continuous interior penalty on the interelement normal gradient jumps as a specific example of symmetric stabilization. The advection and stabilization operators are treated explicitly and the diffusion operator implicitly.

The main result of the paper is to obtain stability and error estimates for smooth solutions. The results are formulated in terms of the Courant and Péclet numbers. Stability and convergence are proved for both cases of advection-dominated and diffusion-dominated regimes.

Finally, two numerical experiments using FreeFem++ are presented to illustrate the analysis, namely, convergence rates to a known smooth solution and control spurious oscillations for a solution with sharp layers.

Reviewer: Angela Handlovičová (Bratislava)

##### MSC:

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35K20 | Initial-boundary value problems for second-order parabolic equations |