An adaptive algorithm for the transport equation with time dependent velocity

An a posteriori error estimate is derived for the approximation of the transport equation with a time dependent transport velocity. Continuous, piecewise linear, anisotropic finite elements are used for space discretization, the Crank-Nicolson scheme scheme is proposed for time discretization. This paper is a generalization of Dubuis S, Picasso M (J Sci Comput 75(1):350-375, 2018) where the transport velocity was not depending on time. The a posteriori error estimate (upper bound) is shown to be sharp for anisotropic meshes, the involved constant being independent of the mesh aspect ratio. A quadratic reconstruction of the numerical solution is introduced in order to obtain an estimate that is order two in time. Error indicators corresponding to space and time are proposed, their accuracy is checked with non-adapted meshes and constant time steps. Then, an adaptive algorithm is introduced, allowing to adapt the meshes and time steps. Numerical experiments are presented when the exact solution has strong variations in space and time, illustrating the efficiency of the method. They indicate that the effectivity index is close to one and does not depend on the solution, mesh size, aspect ratio, and time step.