Volume 22, Issue 2

We propose a duality based a posteriori error estimator for the computation of functionals averaged in time for nonlinear time dependent problems. Such functionals are typically relevant for (quasi-)periodic solutions in time. Applications arise, e.g. in chemical reaction models. In order to reduce the numerical complexity, we use simultaneously locally refined meshes and adaptive (chemical) models. Hence, considerations of adjoint problems measuring the sensitivity of the functional output are needed. In contrast to the classical dual-weighted residual (DWR) method, we favor a fixed mesh and model strategy in time. Taking advantage of the (quasi-)periodic behaviour, only stationary dual problems have to be solved.

The coupled Stokes and Darcy equations are approximated by a strongly conservative finite element method. The discrete spaces are the divergence-conforming velocity space with matching pressure space such as the Raviart-Thomas spaces. This work proves optimal error estimate of the velocity in the L 2 norm in the domain and on the interface. Lipschitz regularity of the interface is sufficient to obtain the results.

In this paper, the previously described monolithic approach [6] for the stationary discrete Boltzmann equation is extended to time-dependent problems. In general, both collision and advection operators are discretized on nonuniform grids as opposed to the standard Lattice Boltzmann method. Implicit time-stepping schemes are applied for an accurate and robust numerical treatment of the nonstationary flow problems. The resulting coupled system of equations is treated using special numerical methods for PDE’s. As in the steady case, we apply a full Newton method for the nonlinear problems, but we also discuss possible variants of semi-implicit schemes all of which lead to nonsymmetric linear systems. The preconditioning of the used Krylov-space methods, resp., the construction of corresponding smoothing operators in the applied multigrid approaches is closely connected to the underlying short characteristic-upwinding discretization, yielding the exact inverse of the transport operators even for unstructured meshes due to a special numbering technique. Numerical results are given for the proposed solvers analysing the efficiency depending on the Mach number, time step and mesh size, while accuracy and stability of the complete space-time discretization are demonstrated for prototypical flow configurations at various timesteps.

This paper considers a special subclass of second derivative general linear methods (SD-GLM). This new class of methods will be known as the second derivative GLMwith nearly almost Runge-Kutta stability (SD-NARK-stability). It is one of the possible generalization of the classical second derivative Runge-Kutta method (SD-RKM) suitable for the solution of non-stiff initial value problems (IVPs) in ordinary differential equations (ODEs).