An ADER-type scheme for a class of equations arising from the water-wave theory

In this work we propose a numerical strategy to solve a family of partial differential equations arising from the water-wave theory. These problems may contain four terms; a source which is an algebraic function of the solution, a convective part involving first order spatial derivatives of the solution, a diffusive part involving second order spatial derivatives and the transient part. Unlike partial differential equations of hyperbolic or parabolic type, where the transient part is the time derivative of the solution, here the transient part can contain mixed time and space derivatives.In [Zambra et al. International Journal for Numerical Methods in Engineering 89(2):227-240, 2012], the authors proposed a globally implicit strategy to solve the Richards equation. In that case, transient terms consisted of algebraic expressions of the solution. Motivated by this work, we propose a one-step finite volume method to deal with problems in which transient terms are differential operators. Here, a locally implicit formulation is investigated, which is based on the ADER philosophy. The scheme is divided in three steps: i) a polynomial reconstruction of the data; ii) solutions to Generalized Riemann Problems (GRP); iii) the solution of differential problems. Note that steps i) and ii), are those of conventional ADER schemes for conservation laws. Advantages of the present approach include the possibility to construct high-order approximations in both space and time, for which existing methodologies for hyperbolic problems can be applied. The differential problems associated to the transient term can be non-linear and numerical strategies can be adopted to deal wit it. Convergence of the scheme is proved rigorously and an empirical convergence rates assessment is carried out in order to illustrate the high space and time accuracy of the present scheme