Monotonicity Conditions for Discretization of Parabolic Conservation Laws

Abstract

In the recent years monotonicity of control volume methods for elliptic equations has been studied. A discrete maximum principle is established in Keilegavlen et al. [18], and a set of monotonicity conditions on general quadrilateral grids has been derived in Nordbotten et al. [23]. Monotonicity criteria for parabolic equations have not yet been studied. We will therefore in this thesis extend the already existing monotonicity conditions for elliptic equations to a set of conditions for parabolic equations. These conditions is derived under the assumption that the discrete maximum principle for parabolic equations is the same as the principle for elliptic problem. It turns out that these conditions are stricter than the elliptic conditions. Since the maximum principle for the time discrete parabolic equation is different from the principle for the elliptic equation, it may be necessary to reformulate the discrete maximum principle. It is not obvious how this shall be done. We will therefore discuss various formulations of time discrete maximum principles together with numerical examples.