Unconditionally stable numerical scheme for the 2D transport equation

The main goal of this paper is to extend the numerical scheme for the transport equation described in previous works [Penel, 2012; Bernard et al., 2014] from one to two dimensional problems. It is based on the method of characteristics, which consists in solving two ordinary differential equations rather than a partial differential equation. Our scheme uses an adaptive 6-point stencil in order to reach second-order accuracy whenever it is possible, and preserves some essential physical properties of the equation, such as the maximum principle. The resulting scheme is proved to be unconditionally stable and to reach second-order accuracy. We show numerical examples with comparisons to the well known Essentially Non-Oscillatory (ENO) scheme [Shu, 1998], in order to illustrate the good properties of our scheme (order of convergence, unconditional stability, accuracy). Using a Gaussian initial condition, several test cases are considered, using a constant or a rotating velocity field, taking into account or not variable source terms. Also, a test is given that shows the possibility of applying the scheme in more realistic fluid mechanics case.